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diff --git a/fvv2/n.lyx b/fvv2/n.lyx
new file mode 100644
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--- /dev/null
+++ b/fvv2/n.lyx
@@ -0,0 +1,218 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
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+\index Index
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+\end_index
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+\end_header
+
+\begin_body
+
+\begin_layout Title
+Funciones de varias variables II
+\end_layout
+
+\begin_layout Date
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+def
+\backslash
+cryear{2018}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "../license.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Bibliografía:
+\end_layout
+
+\begin_layout Itemize
+Funciones de varias variables II: Integración, Antonio José Pallarés Ruiz,
+ Universidad de Murcia (Curso 2010–11).
+\end_layout
+
+\begin_layout Itemize
+Measure and Integral: An Introduction to Real Analysis, Richard L.
+ Wheeden & Antoni Zygmund.
+\end_layout
+
+\begin_layout Itemize
+Cálculo en variedades, Michael Spivak.
+\end_layout
+
+\begin_layout Itemize
+Medida e Integración: Notas de clase para el curso 1998–1999, Antonio José
+ Pallarés Ruiz, Departamento de Matemáticas, Universidad de Murcia.
+\end_layout
+
+\begin_layout Chapter
+Integración Riemann en varias variables
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n1.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Medidas de conjuntos
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset CommandInset include
+LatexCommand input
+filename "n2.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Integración Lebesgue
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n3.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Cambio de variable
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset include
+LatexCommand input
+filename "n4.lyx"
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/fvv2/n1.lyx b/fvv2/n1.lyx
new file mode 100644
index 0000000..0537932
--- /dev/null
+++ b/fvv2/n1.lyx
@@ -0,0 +1,3333 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
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+\fontencoding global
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+\use_package amsmath 1
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+\use_package mathdots 1
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+\use_package mhchem 1
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+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{FUVR2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+partición
+\series default
+ de 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ es una colección de puntos 
+\begin_inset Formula $a=t_{0}<t_{1}<\dots<t_{n}=b$
+\end_inset
+
+ [...].
+ [...]
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $f\in{\cal R}[a,b]$
+\end_inset
+
+, llamamos 
+\series bold
+integral indefinida
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ a la función 
+\begin_inset Formula $F:[a,b]\rightarrow\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $F(x):=\int_{a}^{x}f$
+\end_inset
+
+.
+ El 
+\series bold
+TEOREMA FUNDAMENTAL DEL CÁLCULO
+\series default
+ afirma que, si 
+\begin_inset Formula $f\in{\cal R}[a,b]$
+\end_inset
+
+ y 
+\begin_inset Formula $F$
+\end_inset
+
+ es su integral indefinida, entonces 
+\begin_inset Formula $F$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ y si 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $c\in(a,b)$
+\end_inset
+
+ entonces 
+\begin_inset Formula $F$
+\end_inset
+
+ es derivable en 
+\begin_inset Formula $c$
+\end_inset
+
+ y 
+\begin_inset Formula $F'(c)=f(c)$
+\end_inset
+
+, y esto también ocurre con los extremos del intervalo y las correspondientes
+ derivadas laterales.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Integral de Riemann para funciones de varias variables
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+gráfica
+\series default
+ de una función 
+\begin_inset Formula $f:[a_{1},b_{1}]\times\dots\times[a_{n},b_{n}]\rightarrow\mathbb{R}$
+\end_inset
+
+ a
+\begin_inset Formula 
+\[
+\text{graf}(f):=\{(x_{1},\dots,x_{n},y)\in\mathbb{R}^{n+1}:(x_{1},\dots,x_{n})\in[a_{1},b_{1}]\times\dots\times[a_{n},b_{n}]\land y=f(x_{1},\dots,x_{n})\}
+\]
+
+\end_inset
+
+y 
+\series bold
+subgrafo
+\series default
+ a
+\begin_inset Formula 
+\begin{multline*}
+\text{subgraf}(f):=\\
+\{(x_{1},\dots,x_{n},y)\in\mathbb{R}^{n+1}:(x_{1},\dots,x_{n})\in[a_{1},b_{1}]\times\dots\times[a_{n},b_{n}]\land0\leq y\leq f(x_{1},\dots,x_{n})\}
+\end{multline*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+volumen
+\series default
+ del 
+\series bold
+rectángulo
+\series default
+ 
+\begin_inset Formula $n$
+\end_inset
+
+-dimensional 
+\begin_inset Formula $R:=[a_{1},b_{1}]\times\dots\times[a_{n},b_{n}]\subset\mathbb{R}^{n}$
+\end_inset
+
+ se define como 
+\begin_inset Formula $v(R)=(b_{1}-a_{1})\cdots(b_{n}-a_{n})$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $n=1$
+\end_inset
+
+, 
+\begin_inset Formula $R$
+\end_inset
+
+ es un intervalo y 
+\begin_inset Formula $v(R)$
+\end_inset
+
+ es su 
+\series bold
+longitud
+\series default
+; si 
+\begin_inset Formula $n=2$
+\end_inset
+
+, 
+\begin_inset Formula $R$
+\end_inset
+
+ es un rectángulo y 
+\begin_inset Formula $v(R)$
+\end_inset
+
+ es su 
+\series bold
+área
+\series default
+, y si 
+\begin_inset Formula $n=3$
+\end_inset
+
+, 
+\begin_inset Formula $R$
+\end_inset
+
+ es un paralelepípedo recto y 
+\begin_inset Formula $v(R)$
+\end_inset
+
+ es su 
+\series bold
+volumen
+\series default
+ tridimensional.
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+partición
+\series default
+ sobre este rectángulo es una lista 
+\begin_inset Formula $P:=(P_{1},\dots,P_{n})$
+\end_inset
+
+ en la que cada 
+\begin_inset Formula $P_{i}$
+\end_inset
+
+ es una partición de 
+\begin_inset Formula $[a_{i},b_{i}]$
+\end_inset
+
+.
+ Si cada 
+\begin_inset Formula $P_{i}$
+\end_inset
+
+ divide el intervalo 
+\begin_inset Formula $[a_{i},b_{i}]$
+\end_inset
+
+ en 
+\begin_inset Formula $k_{i}$
+\end_inset
+
+ subintervalos, los 
+\begin_inset Formula $k_{1}\cdots k_{n}$
+\end_inset
+
+ rectángulos en los que 
+\begin_inset Formula $P$
+\end_inset
+
+ divide a 
+\begin_inset Formula $R$
+\end_inset
+
+ son los 
+\series bold
+subrectángulos
+\series default
+ de la partición 
+\begin_inset Formula $P$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f:R\rightarrow\mathbb{R}$
+\end_inset
+
+ es acotada y 
+\begin_inset Formula $P:=(P_{i})_{i=1}^{n}$
+\end_inset
+
+ es una partición de 
+\begin_inset Formula $R$
+\end_inset
+
+ que lo divide en una cantidad finita de subrectángulos 
+\begin_inset Formula $\{S_{i}\}_{i=1}^{N}$
+\end_inset
+
+, si para cada 
+\begin_inset Formula $S_{h}$
+\end_inset
+
+ denotamos 
+\begin_inset Formula $m_{S_{h}}(f):=\inf\{f(x)\}_{x\in S_{h}}$
+\end_inset
+
+ y 
+\begin_inset Formula $M_{S_{h}}(f)=\sup\{f(x)\}_{x\in S_{h}}$
+\end_inset
+
+, y definimos las 
+\series bold
+sumas inferior
+\series default
+ y 
+\series bold
+superior
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ correspondientes a la partición 
+\begin_inset Formula $P$
+\end_inset
+
+, respectivamente, como
+\begin_inset Formula 
+\begin{eqnarray*}
+s(f,P):=\sum_{h=1}^{N}m_{S_{h}}(f)v(S_{h}) & \text{y} & S(f,P):=\sum_{h=1}^{N}M_{S_{h}}(f)v(S_{h})
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $P$
+\end_inset
+
+ y 
+\begin_inset Formula $P'$
+\end_inset
+
+ dos particiones del rectángulo 
+\begin_inset Formula $n$
+\end_inset
+
+-dimensional 
+\begin_inset Formula $R$
+\end_inset
+
+ tales que 
+\begin_inset Formula $P'\succeq P$
+\end_inset
+
+ (
+\begin_inset Formula $\forall i\in\{1,\dots,n\},P'_{i}\succeq P_{i}$
+\end_inset
+
+ o, equivalentemente, cada subrectángulo de 
+\begin_inset Formula $P'$
+\end_inset
+
+ está contenido en uno de 
+\begin_inset Formula $P$
+\end_inset
+
+) y 
+\begin_inset Formula $f:R\rightarrow\mathbb{R}$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+s(f,P)\leq s(f,P')\leq S(f,P')\leq S(f,P)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Obviamente 
+\begin_inset Formula $s(f,P')\leq S(f,P')$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $P$
+\end_inset
+
+ se divide en los subrectángulos 
+\begin_inset Formula $\{S_{i},\dots,S_{N}\}$
+\end_inset
+
+ y 
+\begin_inset Formula $P'$
+\end_inset
+
+ en 
+\begin_inset Formula $\{S'_{1},\dots,S'_{M}\}$
+\end_inset
+
+, dado un subrectángulo 
+\begin_inset Formula $S_{i}$
+\end_inset
+
+ de 
+\begin_inset Formula $P$
+\end_inset
+
+ que se expresa como unión de subrectángulos 
+\begin_inset Formula $S_{i1},\dots,S_{ik_{i}}$
+\end_inset
+
+ de 
+\begin_inset Formula $P'$
+\end_inset
+
+, es claro que para 
+\begin_inset Formula $j\in\{1,\dots,k_{i}\}$
+\end_inset
+
+, 
+\begin_inset Formula $m_{S}(f)=\inf\{f(x)\}_{x\in S_{i}}\leq m_{S_{ij}}(f)=\inf\{f(x)\}_{x\in S_{ij}}$
+\end_inset
+
+.
+ Por otro lado, 
+\begin_inset Formula $v(S_{i})=\sum_{j=1}^{k_{i}}v(S_{ij})$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $m_{S_{i}}(f)v(S_{i})=m_{S_{i}}(f)\sum_{j=1}^{k_{i}}v(S_{ij})=\sum_{j=1}^{k_{i}}m_{S_{i}}(f)v(S_{ij})\leq\sum_{j=1}^{k_{i}}m_{S_{ij}}(f)v(S_{ij})$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $s(f,P)=\sum_{i=1}^{N}m_{S_{i}}(f)v(S_{i})\leq\sum_{i=1}^{N}\sum_{j=1}^{k_{i}}m_{S_{ij}}(f)v(S_{ij})=\sum_{i=1}^{M}m_{S'_{i}}(f)v(S'_{i})=s(f,P')$
+\end_inset
+
+.
+ La prueba de que 
+\begin_inset Formula $S(f,P')\leq S(f,P)$
+\end_inset
+
+ se hace de forma análoga.
+\end_layout
+
+\begin_layout Standard
+Definimos las 
+\series bold
+integrales superior e inferior de Riemann
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $R$
+\end_inset
+
+, respectivamente, como
+\begin_inset Formula 
+\begin{eqnarray*}
+\overline{\int_{R}}f(x_{1},\dots,x_{n})dx_{1}\dots dx_{n} & := & \inf\{S(f,P)\}_{P\text{ partición de }R}\\
+\underline{\int_{R}}f(x_{1},\dots,x_{n})dx_{1}\dots dx_{n} & := & \sup\{s(f,P)\}_{P\text{ partición de }R}
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Por lo anterior, es claro que la integral inferior es siempre menor o igual
+ a la su
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+pe
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+rior.
+ Si son iguales decimos que 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\series bold
+integrable Riemann
+\series default
+ en 
+\begin_inset Formula $R$
+\end_inset
+
+ (
+\begin_inset Formula $f\in{\cal R}(R)$
+\end_inset
+
+) con 
+\series bold
+integral
+\series default
+ 
+\begin_inset Formula $\int_{R}f(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}$
+\end_inset
+
+ igual a estas dos.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f\in{\cal R}(R)\iff\forall\varepsilon>0,\exists P_{\varepsilon}:S(f,P_{\varepsilon})-s(f,P_{\varepsilon})\leq\varepsilon$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{sloppypar}
+\end_layout
+
+\end_inset
+
+Basta tomar 
+\begin_inset Formula $P$
+\end_inset
+
+ y 
+\begin_inset Formula $P'$
+\end_inset
+
+ con 
+\begin_inset Formula $S(f,P)\leq\int_{R}f(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}+\frac{\varepsilon}{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $s(f,P')\geq\int_{R}f(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}-\frac{\varepsilon}{2}$
+\end_inset
+
+ y quedarnos con la partición 
+\begin_inset Formula $P\lor P'$
+\end_inset
+
+ (
+\begin_inset Formula $\{P_{i}\lor P'_{i}\}_{i=1}^{n}$
+\end_inset
+
+).
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{sloppypar}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Si se cumple 
+\begin_inset Formula $0\leq\overline{\int_{R}}f(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}-\underline{\int_{R}}f(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}\leq S(f,P_{\varepsilon})-s(f,P_{\varepsilon})\leq\varepsilon$
+\end_inset
+
+ para todo 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, haciendo tender 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+ a 0 obtenemos la igualdad de las integrales superior e inferior.
+\end_layout
+
+\begin_layout Standard
+La integral es un operador lineal: sean 
+\begin_inset Formula $f,g\in{\cal R}(R)$
+\end_inset
+
+ y 
+\begin_inset Formula $c\in\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $f+g\in{\cal R}(R)$
+\end_inset
+
+ con 
+\begin_inset Formula $\int_{R}(f+g)(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}=\int_{R}f(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}+\int_{R}g(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}$
+\end_inset
+
+, y 
+\begin_inset Formula $cf\in{\cal R}(R)$
+\end_inset
+
+ con 
+\begin_inset Formula $\int_{R}(cf)(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}=c\int_{R}f(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Se deriva de que para 
+\begin_inset Formula $S\subseteq R$
+\end_inset
+
+, 
+\begin_inset Formula $m_{S}(f)+m_{S}(g)\leq m_{S}(f+g)$
+\end_inset
+
+, 
+\begin_inset Formula $M_{S}(f+g)\leq M_{S}(f)+M_{S}(g)$
+\end_inset
+
+ y 
+\begin_inset Formula $M_{S}(cf)=cM_{S}(f)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Newpage newpage
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+También es un operador positivo.
+ Sean 
+\begin_inset Formula $f,g\in{\cal R}(R)$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall(x_{1},\dots,x_{n})\in R,f(x_{1},\dots,x_{n})\geq0\implies\int_{R}(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}\geq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall(x_{1},\dots,x_{n})\in R,f(x_{1},\dots,x_{n})\geq g(x_{1},\dots,x_{n})\implies\int_{R}f(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}\geq\int_{R}g(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $|f|\in{\cal R}(R)$
+\end_inset
+
+ y 
+\begin_inset Formula $\left|\int_{R}f(x_{1},\dots,x_{n})dx_{1}\dots dx_{n}\right|\leq\int_{R}|f(x_{1},\dots,x_{n})|dx_{1}\dots dx_{n}$
+\end_inset
+
+ (desigualdad triangular).
+\end_layout
+
+\begin_layout Standard
+Definimos la 
+\series bold
+oscilación
+\series default
+ de 
+\begin_inset Formula $f:R\rightarrow\mathbb{R}$
+\end_inset
+
+ en 
+\begin_inset Formula $S\subseteq R$
+\end_inset
+
+ como
+\begin_inset Formula 
+\[
+\text{osc}(f,S):=M_{S}(f)-m_{S}(f)=\sup\{|f(x)-f(y)|\}_{x,y\in S}
+\]
+
+\end_inset
+
+ y la oscilación de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $x\in R$
+\end_inset
+
+ como
+\begin_inset Formula 
+\[
+\text{osc}(f,x):=\inf\{\text{osc}(f,S\cap T)\}_{T\text{ rectángulo abierto centrado en }S}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vemos que 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $x$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $\text{osc}(f,x)=0$
+\end_inset
+
+, y que para cada 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+ y cada 
+\begin_inset Formula $x$
+\end_inset
+
+ donde 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua existe un cubo abierto 
+\begin_inset Formula $C_{x}(d_{x})$
+\end_inset
+
+ centrado en 
+\begin_inset Formula $x$
+\end_inset
+
+ con diámetro 
+\begin_inset Formula $d_{x}$
+\end_inset
+
+ donde la oscilación de 
+\begin_inset Formula $f$
+\end_inset
+
+ es menor que 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{TEM}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Toda 
+\begin_inset Formula $f:(X,d)\rightarrow(Y,d')$
+\end_inset
+
+ continua, siendo 
+\begin_inset Formula $(X,{\cal T}_{d})$
+\end_inset
+
+ compacto, es uniformemente continua.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, toda 
+\begin_inset Formula $f:R\rightarrow\mathbb{R}$
+\end_inset
+
+ continua definida en un rectángulo cerrado 
+\begin_inset Formula $n$
+\end_inset
+
+-dimensional 
+\begin_inset Formula $R$
+\end_inset
+
+ es integrable Riemann en 
+\begin_inset Formula $R$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Conjuntos de contenido y de medida nula
+\end_layout
+
+\begin_layout Standard
+Un subconjunto 
+\begin_inset Formula $S\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ tiene 
+\series bold
+medida
+\series default
+ (
+\begin_inset Formula $n$
+\end_inset
+
+-dimensional) 
+\series bold
+nula
+\series default
+ si para cada 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+ existe una sucesión 
+\begin_inset Formula $(R_{k})_{k}$
+\end_inset
+
+ de rectángulos 
+\begin_inset Formula $n$
+\end_inset
+
+-dimensionales cerrados tal que 
+\begin_inset Formula $S\subseteq\bigcup_{j=1}^{\infty}R_{j}$
+\end_inset
+
+ y 
+\begin_inset Formula $\sum_{j=1}^{\infty}v(R_{j})<\varepsilon$
+\end_inset
+
+.
+ Sustituyendo los rectángulos cerrados por rectángulos abiertos obtenemos
+ el mismo concepto.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Fijado 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+, sea 
+\begin_inset Formula $(R_{k})_{k}$
+\end_inset
+
+ la sucesión de cerrados que cumple las propiedades y 
+\begin_inset Formula $0<d<\varepsilon-\sum_{j=1}^{\infty}v(R_{j})$
+\end_inset
+
+.
+ Para cada rectángulo 
+\begin_inset Formula $R_{k}$
+\end_inset
+
+, tomamos un rectángulo abierto 
+\begin_inset Formula $R'_{k}\supseteq R_{k}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $v(R'_{k})\leq R_{k}+\frac{d}{2^{k+1}}$
+\end_inset
+
+, y vemos que, en efecto, 
+\begin_inset Formula $\sum_{j=1}^{\infty}v(R'_{j})\leq\sum_{j=1}^{\infty}v(R_{j})+\sum_{j=1}^{\infty}\frac{d}{2^{k+1}}<\varepsilon$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Fijado 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, sea 
+\begin_inset Formula $(R_{k})_{k\in\mathbb{N}}$
+\end_inset
+
+ una sucesión de rectángulos abiertos que cumple las propiedades, vemos
+ que 
+\begin_inset Formula $N\subseteq\bigcup_{k\in\mathbb{N}}R_{k}\subseteq\bigcup_{k\in\mathbb{N}}\overline{R_{k}}$
+\end_inset
+
+ y que 
+\begin_inset Formula $\sum_{j=1}^{\infty}v(\overline{R_{j}})=\sum_{j=1}^{\infty}v(R_{j})<\varepsilon$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, la unión numerable de conjuntos de medida nula tiene medida nula.
+ 
+\series bold
+Demostración:
+\series default
+ Consideremos 
+\begin_inset Formula $\bigcup_{i=1}^{\infty}S_{i}$
+\end_inset
+
+.
+ Fijado 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+, para cada 
+\begin_inset Formula $i\in\mathbb{N}$
+\end_inset
+
+ utilizamos que 
+\begin_inset Formula $S_{i}$
+\end_inset
+
+ es de medida nula para recubrirlo con una sucesión 
+\begin_inset Formula $\{R_{ij}\}_{j\in\mathbb{N}}$
+\end_inset
+
+ cuyos volúmenes suman menos que 
+\begin_inset Formula $\frac{\varepsilon}{2^{i}}$
+\end_inset
+
+.
+ Vemos que 
+\begin_inset Formula $\{R_{ij}\}_{i,j\in\mathbb{N}}$
+\end_inset
+
+ es numerable y podemos describirlo como una sucesión que recubre 
+\begin_inset Formula $N$
+\end_inset
+
+ y cuyos volúmenes suman menos que 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+, pues 
+\begin_inset Formula $\sum_{i,j=1}^{\infty}v(R_{ij})=\sum_{i=1}^{\infty}(\sum_{j=1}^{\infty}v(R_{ij}))<\sum_{i=1}^{\infty}\frac{\varepsilon}{2^{i}}=\varepsilon$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Un subconjunto 
+\begin_inset Formula $S\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ tiene 
+\series bold
+contenido
+\series default
+ (
+\begin_inset Formula $n$
+\end_inset
+
+-dimensional) 
+\series bold
+nulo
+\series default
+ si para cada 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+ existe una familia finita de rectángulos 
+\begin_inset Formula $n$
+\end_inset
+
+-dimensionales cerrados que cumplen las mismas condiciones que los de la
+ definición de medida nula.
+ Es claro que todo conjunto de contenido nulo tiene medida nula.
+ Como 
+\series bold
+teorema
+\series default
+, todo compacto en 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ de medida nula es de contenido nulo; basta usar un cubrimiento con rectángulos
+ abiertos en la definición de medida nula y extraer un subrecubrimiento
+ finito por compacidad.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f:R\subseteq\mathbb{R}^{m}\rightarrow\mathbb{R}$
+\end_inset
+
+ es integrable Riemann en el rectángulo cerrado 
+\begin_inset Formula $R$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\text{graf}(f)$
+\end_inset
+
+ tiene contenido 
+\begin_inset Formula $(m+1)$
+\end_inset
+
+-dimensional nulo.
+\end_layout
+
+\begin_layout Subsection
+El conjunto de Cantor
+\end_layout
+
+\begin_layout Standard
+Consideremos el intervalo 
+\begin_inset Formula $[0,1]$
+\end_inset
+
+, que dividimos en 3 subintervalos y eliminamos el subintervalo abierto
+ central, 
+\begin_inset Formula $(\frac{1}{3},\frac{2}{3})$
+\end_inset
+
+.
+ A continuación, de cada subintervalo cerrado restante, eliminamos el subinterva
+lo abierto central de longitud la tercera parte del subintervalo original.
+ Repitiendo este proceso indefinidamente lo que nos queda es el 
+\series bold
+conjunto de Cantor
+\series default
+, 
+\begin_inset Formula $C$
+\end_inset
+
+.
+ Observamos que un número está en el conjunto de Cantor si y sólo si su
+ representación en base 3, 
+\begin_inset Formula $0.c_{1}c_{2}c_{3}\cdots c_{n}\cdots$
+\end_inset
+
+, contiene sólo los dígitos 0 y 2, teniendo en cuenta que el número puede
+ también acabar por una secuencia infinita de doses.
+ Teoremas:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $C$
+\end_inset
+
+ es incontable, con igual cardinalidad que 
+\begin_inset Formula $[0,1]$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+La función 
+\begin_inset Formula $f:C\rightarrow[0,1]$
+\end_inset
+
+ dada por 
+\begin_inset Formula $f(0.c_{1}c_{2}\cdots c_{n}\cdots_{(3)}):=0.\frac{c_{1}}{2}\frac{c_{2}}{2}\cdots\frac{c_{n}}{2}\cdots_{(2)}$
+\end_inset
+
+ es suprayectiva, luego 
+\begin_inset Formula $|C|\geq|[0,1]|$
+\end_inset
+
+, pero es claro que 
+\begin_inset Formula $|C|\leq|[0,1]|$
+\end_inset
+
+, luego 
+\begin_inset Formula $|C|=|[0,1]|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $C$
+\end_inset
+
+ es de medida nula.
+\begin_inset Newline newline
+\end_inset
+
+Al principio 
+\begin_inset Formula $[0,1]$
+\end_inset
+
+ tiene longitud 1, y es fácil ver que si en un 
+\begin_inset Quotes fld
+\end_inset
+
+paso
+\begin_inset Quotes frd
+\end_inset
+
+ de la construcción el conjunto resultante tiene longitud 
+\begin_inset Formula $n$
+\end_inset
+
+, en el siguiente tendrá longitud 
+\begin_inset Formula $\frac{2}{3}n$
+\end_inset
+
+.
+ Por tanto la longitud de 
+\begin_inset Formula $C$
+\end_inset
+
+ es 
+\begin_inset Formula $\lim_{n}\left(\frac{2}{3}\right)^{n}=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $C$
+\end_inset
+
+ no tiene puntos interiores.
+\begin_inset Newline newline
+\end_inset
+
+Como es de medida nula no contiene puntos de acumulación, pues para ello
+ debería contener intervalos, de medida no nula.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $C$
+\end_inset
+
+ está acotado.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $C$
+\end_inset
+
+ es cerrado.
+\begin_inset Newline newline
+\end_inset
+
+Es el resultado de quitar a un cerrado (
+\begin_inset Formula $[0,1]$
+\end_inset
+
+) un abierto (la unión de los abiertos eliminados en su construcción).
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $C$
+\end_inset
+
+ no tiene puntos aislados.
+\begin_inset Newline newline
+\end_inset
+
+Dado 
+\begin_inset Formula $x\in C$
+\end_inset
+
+ y 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, sea 
+\begin_inset Formula $n$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\frac{2}{3^{n}}<\varepsilon$
+\end_inset
+
+, existe un punto, el resultado de cambiar la cifra 
+\begin_inset Formula $n$
+\end_inset
+
+-ésima de 
+\begin_inset Formula $x$
+\end_inset
+
+ por un 2 si era un 0 o viceversa, cuya distancia a 
+\begin_inset Formula $x$
+\end_inset
+
+ es menor que 
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Dados 
+\begin_inset Formula $a,b\in C$
+\end_inset
+
+ distintos, existe una partición 
+\begin_inset Formula $\{A,B\}$
+\end_inset
+
+ de 
+\begin_inset Formula $C$
+\end_inset
+
+ con 
+\begin_inset Formula $A$
+\end_inset
+
+ y 
+\begin_inset Formula $B$
+\end_inset
+
+ cerrados, 
+\begin_inset Formula $a\in A$
+\end_inset
+
+ y 
+\begin_inset Formula $b\in B$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Sea 
+\begin_inset Formula $n$
+\end_inset
+
+ la posición de una cifra (en base 3) en la que 
+\begin_inset Formula $a$
+\end_inset
+
+ y 
+\begin_inset Formula $b$
+\end_inset
+
+ difieren, basta hacer la partición según el valor de dicha cifra.
+\end_layout
+
+\begin_layout Subsection
+Caracterización
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $A\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ cerrado, si 
+\begin_inset Formula $f:A\rightarrow\mathbb{R}$
+\end_inset
+
+ es acotada y 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, 
+\begin_inset Formula $B:=\{x\in A:\text{osc}(f,x)\geq\varepsilon\}$
+\end_inset
+
+ es cerrado.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $x\in\mathbb{R}^{n}\backslash B$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $x\notin A$
+\end_inset
+
+, existe una bola de centro 
+\begin_inset Formula $x$
+\end_inset
+
+ que no interseca con 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $x\in A$
+\end_inset
+
+, existe un rectángulo abierto 
+\begin_inset Formula $C\ni x$
+\end_inset
+
+ con 
+\begin_inset Formula $\text{osc}(f,c)<\varepsilon$
+\end_inset
+
+, y para 
+\begin_inset Formula $y\in C$
+\end_inset
+
+ existe 
+\begin_inset Formula $\delta_{y}$
+\end_inset
+
+ tal que si 
+\begin_inset Formula $\Vert z-y\Vert<\delta_{y}$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\Vert x-z\Vert<\delta$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\text{osc}(f,y)<\varepsilon$
+\end_inset
+
+ y 
+\begin_inset Formula $C\subseteq\mathbb{R}^{n}\backslash B$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+teorema de Lebesgue de caracterización de las funciones integrables
+\series default
+ afirma que si 
+\begin_inset Formula $R$
+\end_inset
+
+ es un rectángulo 
+\begin_inset Formula $n$
+\end_inset
+
+-dimensional cerrado y 
+\begin_inset Formula $f:R\rightarrow\mathbb{R}$
+\end_inset
+
+ es acotada, entonces 
+\begin_inset Formula $f\in{\cal R}(R)$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $B:=\{x\in R:f\text{ no es continua en }x\}$
+\end_inset
+
+ tiene medida nula.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $B_{k}:=\{x\in R:o(f,x)\geq\frac{1}{k}\}$
+\end_inset
+
+, basta probar que cada 
+\begin_inset Formula $B_{k}$
+\end_inset
+
+ tiene medida nula dado que 
+\begin_inset Formula $B=\bigcup_{k\in\mathbb{N}}B_{k}$
+\end_inset
+
+.
+ Dado 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, sean 
+\begin_inset Formula $P$
+\end_inset
+
+ una partición de 
+\begin_inset Formula $R$
+\end_inset
+
+ con 
+\begin_inset Formula $S(f,P)-s(f,P)<\frac{\varepsilon}{k}$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal S}$
+\end_inset
+
+ el conjunto de subrectángulos de 
+\begin_inset Formula $P$
+\end_inset
+
+ que cortan a 
+\begin_inset Formula $B_{k}$
+\end_inset
+
+ , entonces para 
+\begin_inset Formula $S\in{\cal S}$
+\end_inset
+
+ se tiene 
+\begin_inset Formula $M_{S}(f)-m_{S}(f)\geq\frac{1}{k}$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\frac{1}{k}\sum_{S\in{\cal S}}v(S)\leq\sum_{S\in{\cal S}}(M_{S}(f)-m_{S}(f))v(S)\leq S(f,P)-s(f,P)<\frac{\varepsilon}{k}
+\]
+
+\end_inset
+
+con lo que 
+\begin_inset Formula $\sum_{S\in{\cal S}}v(S)<\varepsilon$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Fijado 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, sea 
+\begin_inset Formula $K$
+\end_inset
+
+ tal que 
+\begin_inset Formula $|f(x)|\leq K\forall x\in R$
+\end_inset
+
+, y tenemos que 
+\begin_inset Formula $M_{N}(f)-m_{N}(f)\leq2K\forall N\subseteq R$
+\end_inset
+
+.
+ Consideramos el conjunto de puntos donde 
+\begin_inset Formula $\text{osc}(f,x)\geq\frac{\varepsilon}{2v(R)}$
+\end_inset
+
+, un cerrado de medida nula y por tanto un compacto de contenido nulo, que
+ podemos cubrir con una cantidad finita de abiertos 
+\begin_inset Formula $N_{k}$
+\end_inset
+
+ tales que 
+\begin_inset Formula $\sum_{k}v(N_{k})<\frac{\varepsilon}{4K}$
+\end_inset
+
+, y por tanto 
+\begin_inset Formula $\sum_{k}\text{osc}(N_{k},f)v(N_{k})<2K\cdot\frac{\varepsilon}{4K}=\frac{\varepsilon}{2}$
+\end_inset
+
+.
+ Es claro que 
+\begin_inset Formula $C:=R\backslash\bigcup_{k}N_{k}$
+\end_inset
+
+ es compacto, y como para cada 
+\begin_inset Formula $x\in C$
+\end_inset
+
+ podemos tomar un 
+\begin_inset Formula $S_{x}$
+\end_inset
+
+ abierto tal que 
+\begin_inset Formula $\text{osc}(f,S_{x})<\frac{\varepsilon}{2v(R)}$
+\end_inset
+
+, existe un subrecubrimiento finito 
+\begin_inset Formula $S_{x_{i}}$
+\end_inset
+
+ a partir de este, de modo que 
+\begin_inset Formula $\sum_{i}\text{osc}(f,S_{x_{i}})v(S_{x_{i}})<\frac{\varepsilon}{2v(R)}\cdot v(R)=\frac{\varepsilon}{2}$
+\end_inset
+
+ La partición 
+\begin_inset Formula $P$
+\end_inset
+
+ cuyos subintervalos están contenidos bien en un rectángulo 
+\begin_inset Formula $S_{x_{i}}$
+\end_inset
+
+ o 
+\begin_inset Formula $N_{k}$
+\end_inset
+
+ cumple que 
+\begin_inset Formula $S(f,P)-s(f,P)\leq\sum_{k}\text{osc}(f,N_{k})v(N_{k})+\sum_{i}\text{osc}(f,S_{x_{i}})v(S_{x_{i}})<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Conjuntos medibles Jordan
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ acotado es 
+\series bold
+medible Jordan
+\series default
+ si su 
+\series bold
+función característica
+\series default
+,
+\begin_inset Formula 
+\[
+\chi_{A}(x):=\begin{cases}
+1 & \text{si }x\in A\\
+0 & \text{si }x\notin A
+\end{cases}
+\]
+
+\end_inset
+
+ es integrable Riemann en un rectángulo cerrado 
+\begin_inset Formula $R\supseteq A$
+\end_inset
+
+, y se define el 
+\series bold
+volumen
+\series default
+ 
+\begin_inset Formula $n$
+\end_inset
+
+-dimensional de 
+\begin_inset Formula $A$
+\end_inset
+
+ como
+\begin_inset Formula 
+\[
+v(A):=\int_{R}\chi_{A}(x_{1},\dots,x_{n})dx_{1}\cdots dx_{n}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Equivalentemente, 
+\begin_inset Formula $A$
+\end_inset
+
+ es medible Jordan si y sólo si su frontera, 
+\begin_inset Formula $\partial A=\overline{A}\backslash\mathring{A}$
+\end_inset
+
+, tiene medida nula.
+ Se dice que una función acotada 
+\begin_inset Formula $f:A\rightarrow\mathbb{R}$
+\end_inset
+
+ es integrable Riemann en 
+\begin_inset Formula $A$
+\end_inset
+
+ si 
+\begin_inset Formula $f\chi_{A}\in{\cal R}(R)$
+\end_inset
+
+.
+ Por ejemplo, si 
+\begin_inset Formula $N$
+\end_inset
+
+ tiene contenido nulo y 
+\begin_inset Formula $f:N\rightarrow\mathbb{R}$
+\end_inset
+
+ es acotada, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable Riemann en 
+\begin_inset Formula $N$
+\end_inset
+
+ y 
+\begin_inset Formula $\int_{N}f(x)dx=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Teorema de Fubini
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{sloppypar}
+\end_layout
+
+\end_inset
+
+Sean 
+\begin_inset Formula $R_{1}\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $R_{2}\subseteq\mathbb{R}^{m}$
+\end_inset
+
+ rectángulos cerrados de dimensiones respectivas 
+\begin_inset Formula $n$
+\end_inset
+
+ y 
+\begin_inset Formula $m$
+\end_inset
+
+, 
+\begin_inset Formula $R=R_{1}\times R_{2}\subseteq\mathbb{R}^{n+m}$
+\end_inset
+
+ un rectángulo cerrado 
+\begin_inset Formula $(n+m)$
+\end_inset
+
+-dimensional y 
+\begin_inset Formula $f:R\rightarrow\mathbb{R}$
+\end_inset
+
+ una función acotada.
+ Para cada 
+\begin_inset Formula $x\in R_{1}$
+\end_inset
+
+ definimos 
+\begin_inset Formula $lf_{x}:R_{2}\rightarrow\mathbb{R}$
+\end_inset
+
+ como 
+\begin_inset Formula $lf_{x}(y):=f(x,y)$
+\end_inset
+
+, 
+\begin_inset Formula $s_{lf}(x):=\underline{\int_{R_{2}}}lf_{x}(y_{1},\dots,y_{m})dy_{1}\cdots dy_{m}$
+\end_inset
+
+ y 
+\begin_inset Formula $S_{lf}(x):=\overline{\int_{R_{2}}}lf_{x}(y_{1},\dots,y_{m})dy_{1}\cdots dy_{m}$
+\end_inset
+
+, y para cada 
+\begin_inset Formula $y\in R_{2}$
+\end_inset
+
+ definimos 
+\begin_inset Formula $rf_{y}(x):=f(x,y)$
+\end_inset
+
+, 
+\begin_inset Formula $s_{rf}(y):=\int_{R_{1}}rf_{y}(x_{1},\dots,x_{n})dx_{1}\cdots dx_{m}$
+\end_inset
+
+ y 
+\begin_inset Formula $S_{rf}(y):=\overline{\int_{R_{1}}}rf_{y}(x_{1},\dots,x_{n})dx_{1}\cdots dx_{m}$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $f\in{\cal R}(R)$
+\end_inset
+
+ entonces 
+\begin_inset Formula $s_{lf},S_{lf}\in{\cal R}(R_{1})$
+\end_inset
+
+, 
+\begin_inset Formula $s_{rf},S_{rf}\in{\cal R}(R_{2})$
+\end_inset
+
+ y
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{sloppypar}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula 
+\begin{multline*}
+\int_{R}f(x_{1},\dots,x_{n},y_{1},\dots,y_{m})dx_{1}\cdots dx_{n}dy_{1}\cdots dy_{m}=\\
+=\int_{R_{1}}s_{lf}(x_{1},\dots,x_{n})dx_{1}\cdots dx_{n}=\int_{R_{1}}S_{lf}(x_{1},\dots,x_{n})dx_{1}\cdots dx_{n}=\\
+=\int_{R_{2}}s_{rf}(y_{1},\dots,y_{m})dy_{1}\cdots dy_{m}=\int_{R_{2}}S_{rf}(y_{1},\dots,y_{m})dy_{1}\cdots dy_{m}
+\end{multline*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+En la práctica esto significa que
+\begin_inset Formula 
+\[
+\int_{R}f(\vec{x},\vec{y})d\vec{x}d\vec{y}=\int_{R_{1}}\left(\int_{R_{2}}f(\vec{x},\vec{y})d\vec{y}\right)d\vec{x}=\int_{R_{2}}\left(\int_{R_{1}}f(\vec{x},\vec{y})d\vec{x}\right)d\vec{y}
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula $d\vec{x}:=dx_{1}\cdots dx_{n}$
+\end_inset
+
+ y 
+\begin_inset Formula $d\vec{y}:=dy_{1}\cdots dy_{m}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+sremember{FUVR2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Cálculo de primitivas
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\int u^{n}u'\,dx=\frac{u^{n+1}}{n+1}+C\forall n\neq-1$
+\end_inset
+
+; 
+\begin_inset Formula $\int\frac{u'}{u}dx=\ln|u|+C\forall u\neq0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\int e^{u}u'\,dx=e^{u}+C$
+\end_inset
+
+; 
+\begin_inset Formula $\int a^{u}u'\,dx=\frac{a^{u}}{\ln a}+C\forall a>0,a\neq1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\int\cos u\,u'\,dx=\sin u+C$
+\end_inset
+
+; 
+\begin_inset Formula $\int\sin u\,u'\,dx=-\cos u+C$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\int\cosh u\,u'\,dx=\sinh u+C$
+\end_inset
+
+; 
+\begin_inset Formula $\int\sinh u\,u'\,dx=\cosh u+C$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\int\frac{u'}{\sin^{2}u}dx=\int\frac{u'}{\sinh^{2}u}dx=-\cot u+C$
+\end_inset
+
+; 
+\begin_inset Formula $\int\frac{u'}{\cos^{2}u}dx=\int\frac{u'}{\cosh^{2}u}dx=\tan u+C$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\int\frac{u'}{1+u^{2}}dx=\arctan u+C$
+\end_inset
+
+; 
+\begin_inset Formula $\int\frac{u'}{1-u^{2}}dx=\arg\tanh u+C$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\int\frac{u'}{\sqrt{1-u^{2}}}dx=\arcsin u+C=-\arccos u+C'$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\int\frac{u'}{\sqrt{u^{2}+1}}dx=\arg\sinh u+C$
+\end_inset
+
+; 
+\begin_inset Formula $\int\frac{u'}{\sqrt{u^{2}-1}}dx=\arg\cosh u+C$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\backslash
+sremember{FUVR2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\begin{eqnarray*}
+\cosh(x)=\frac{e^{x}+e^{-x}}{2} &  & \arg\cosh(x)=\ln(x+\sqrt{x^{2}-1})\\
+\sinh(x)=\frac{e^{x}-e^{-x}}{2} &  & \arg\sinh(x)=\ln(x+\sqrt{x^{2}+1})\\
+\cosh^{2}(x)-\sinh^{2}(x)=1 &  & \arg\tanh(x)=\frac{1}{2}\ln\frac{1+x}{1-x}
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Integración por partes
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $f,g\in{\cal R}[a,b]$
+\end_inset
+
+ con primitivas respectivas 
+\begin_inset Formula $F$
+\end_inset
+
+ y 
+\begin_inset Formula $G$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\int_{a}^{b}Fg=F(b)G(b)-F(a)G(a)-\int_{a}^{b}fG
+\]
+
+\end_inset
+
+lo que suele escribirse como 
+\begin_inset Formula $\int u\,dv=uv-\int v\,du$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Cambio de variable
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, sea 
+\begin_inset Formula $\varphi:[c,d]\rightarrow[a,b]\in{\cal C}^{1}[c,d]$
+\end_inset
+
+ con 
+\begin_inset Formula $\varphi(c)=a$
+\end_inset
+
+ y 
+\begin_inset Formula $\varphi(d)=b$
+\end_inset
+
+, sea 
+\begin_inset Formula $f:[a,b]\rightarrow\mathbb{R}$
+\end_inset
+
+ continua, entonces
+\begin_inset Formula 
+\[
+\int_{a}^{b}f=\int_{c}^{d}(f\circ\varphi)\varphi'
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Funciones racionales
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $P(x)$
+\end_inset
+
+ y 
+\begin_inset Formula $Q(x)$
+\end_inset
+
+ polinomios y queremos resolver 
+\begin_inset Formula $\int_{a}^{b}\frac{P(x)}{Q(x)}dx$
+\end_inset
+
+.
+ Si el grado de 
+\begin_inset Formula $P(x)$
+\end_inset
+
+ es mayor o igual que el de 
+\begin_inset Formula $Q(x)$
+\end_inset
+
+ hacemos 
+\begin_inset Formula $\int_{a}^{b}\frac{P(x)}{Q(x)}dx=\int C(x)dx+\int\frac{R(x)}{Q(x)}dx$
+\end_inset
+
+ para que el grado del numerador sea menor que el del denominador.
+ Entonces descomponemos en fracciones simples.
+\end_layout
+
+\begin_layout Standard
+Descomponemos 
+\begin_inset Formula $Q(x)$
+\end_inset
+
+ como 
+\begin_inset Formula $Q(x)=\prod_{i=1}^{r}(x-a_{i})^{m_{i}}\prod_{i=1}^{s}(x^{2}+p_{i}x+q_{i})^{n_{i}}$
+\end_inset
+
+, donde 
+\begin_inset Formula $q_{i}>\frac{p_{i}^{2}}{4}$
+\end_inset
+
+ para que los factores sean irreducibles.
+ Entonces (si el grado de 
+\begin_inset Formula $P(x)$
+\end_inset
+
+ es menor que el de 
+\begin_inset Formula $Q(x)$
+\end_inset
+
+) podemos expresar la fracción como
+\begin_inset Formula 
+\[
+\frac{P(x)}{Q(x)}=\sum_{i=1}^{r}\sum_{j=1}^{m_{i}}\frac{A_{ij}}{(x-a_{i})^{j}}+\sum_{i=1}^{M}\sum_{j=1}^{n_{i}}\frac{M_{ij}x+N_{ij}}{(x^{2}+p_{i}x+q_{i})^{j}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Resolvemos los 
+\begin_inset Formula $A_{k,i}$
+\end_inset
+
+, 
+\begin_inset Formula $M_{k,i}$
+\end_inset
+
+, 
+\begin_inset Formula $N_{k,i}$
+\end_inset
+
+ y nos queda hallar la integral de cada sumando como sigue:
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\int\frac{A}{x-a}dx=A\ln|x-a|+C$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\int\frac{A}{(x-a)^{n}}dx=-\frac{A}{(n-1)(x-a)^{n-1}}+C$
+\end_inset
+
+, donde 
+\begin_inset Formula $n\in2,3,\dots$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\backslash
+sremember{FUVR2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\int\frac{Mx+N}{x^{2}+px+q}dx=\frac{M}{2}\ln\left(\left(x+\frac{p}{2}\right)^{2}+c^{2}\right)+\frac{N-\frac{Mp}{2}}{c}\arctan\left(\frac{x+\frac{p}{2}}{c}\right)+C$
+\end_inset
+
+, donde 
+\begin_inset Formula $c=\frac{\sqrt{4q-p^{2}}}{2}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Funciones que contienen 
+\begin_inset Formula $\cos x$
+\end_inset
+
+ y 
+\begin_inset Formula $\sin x$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+En general, haremos 
+\begin_inset Formula $t=\tan\frac{x}{2}$
+\end_inset
+
+ y entonces
+\begin_inset Formula 
+\begin{eqnarray*}
+\cos x=\frac{\cos(2\frac{x}{2})}{\sin^{2}\frac{x}{2}+\cos^{2}\frac{x}{2}}=\frac{\cos^{2}\frac{x}{2}-\sin^{2}\frac{x}{2}}{\sin^{2}\frac{x}{2}+\cos^{2}\frac{x}{2}} & \overset{\text{div. }\cos^{2}\frac{x}{2}}{=} & \frac{1-\tan^{2}\frac{x}{2}}{\tan^{2}\frac{x}{2}+1}=\frac{1-t^{2}}{1+t^{2}}\\
+\sin x=\frac{\sin(2\frac{x}{2})}{\sin^{2}\frac{x}{2}+\cos^{2}\frac{x}{2}}=\frac{2\sin\frac{x}{2}\cos\frac{x}{2}}{\sin^{2}\frac{x}{2}+\cos^{2}\frac{x}{2}} & \overset{\text{div. }\cos^{2}\frac{x}{2}}{=} & \frac{2\tan\frac{x}{2}}{\tan^{2}\frac{x}{2}+1}=\frac{2t}{1+t^{2}}\\
+x=2\arctan t & \text{ y } & dx=\frac{2}{1+t^{2}}dt
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si la función es de la forma 
+\begin_inset Formula $f(x)=g(\sin x)\cos x$
+\end_inset
+
+, siendo 
+\begin_inset Formula $g$
+\end_inset
+
+ una función racional, hacemos 
+\begin_inset Formula $t=\sin x$
+\end_inset
+
+, y si es 
+\begin_inset Formula $f(x)=g(\cos x)\sin x$
+\end_inset
+
+ hacemos 
+\begin_inset Formula $t=\cos x$
+\end_inset
+
+.
+ Si es 
+\begin_inset Formula $f(x)=g(\tan x)$
+\end_inset
+
+ hacemos 
+\begin_inset Formula $\tan x=t$
+\end_inset
+
+, y podemos llegar a esta situación cuando al sustituir 
+\begin_inset Formula $\sin x$
+\end_inset
+
+ por 
+\begin_inset Formula $\cos x\tan x$
+\end_inset
+
+ quedan solo potencias pares de 
+\begin_inset Formula $\cos x$
+\end_inset
+
+, y hacemos 
+\begin_inset Formula $\cos^{2}x=\frac{1}{1+\tan^{2}x}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+En el caso 
+\begin_inset Formula $f(x)=\cos^{n}x\sin^{m}x$
+\end_inset
+
+, si 
+\begin_inset Formula $n$
+\end_inset
+
+ es impar hacemos 
+\begin_inset Formula $t=\sin x$
+\end_inset
+
+, si 
+\begin_inset Formula $m$
+\end_inset
+
+ es impar, 
+\begin_inset Formula $t=\cos x$
+\end_inset
+
+, y si ambos son pares, usamos 
+\begin_inset Formula $\cos^{2}x=\frac{1+\cos(2x)}{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $\sin^{2}x=\frac{1-\cos(2x)}{2}$
+\end_inset
+
+ para 
+\begin_inset Quotes cld
+\end_inset
+
+reducir el grado
+\begin_inset Quotes crd
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Funciones de la forma 
+\begin_inset Formula $f(e^{x})$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Hacemos el cambio 
+\begin_inset Formula $t=e^{x}$
+\end_inset
+
+ y 
+\begin_inset Formula $dt=e^{x}dx$
+\end_inset
+
+, y esto también sirve para el coseno y seno hiperbólicos (
+\begin_inset Formula $\cosh$
+\end_inset
+
+ y 
+\begin_inset Formula $\sinh$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Subsection
+Funciones que contienen 
+\begin_inset Formula $\sqrt{ax^{2}+2bx+c}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula $d:=\frac{ac-b^{2}}{a}$
+\end_inset
+
+ y se tiene 
+\begin_inset Formula $ax^{2}+2bx+c=a\left(x+\frac{b}{a}\right)^{2}+d$
+\end_inset
+
+.
+ Hacemos entonces el cambio de variable 
+\begin_inset Formula $t=x+\frac{b}{a}$
+\end_inset
+
+ y a continuación:
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $a>0$
+\end_inset
+
+ y 
+\begin_inset Formula $d>0$
+\end_inset
+
+ hacemos 
+\begin_inset Formula $at^{2}=d\tan^{2}u$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\sqrt{at^{2}+d}=[...]\sqrt{d}\sec u$
+\end_inset
+
+ y 
+\begin_inset Formula $dt=\sqrt{\frac{d}{a}}\sec^{2}u\,du$
+\end_inset
+
+.
+ También podemos hacer 
+\begin_inset Formula $at^{2}=d\sinh^{2}u$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\sqrt{at^{2}+d}=[...]\sqrt{d}\cosh u$
+\end_inset
+
+ y 
+\begin_inset Formula $dt=\sqrt{\frac{d}{a}}\cosh u\,du$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $a>0$
+\end_inset
+
+ y 
+\begin_inset Formula $d<0$
+\end_inset
+
+ hacemos 
+\begin_inset Formula $at^{2}=-d\sec^{2}u$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\sqrt{at^{2}+d}=[...]\sqrt{-d}\tan u$
+\end_inset
+
+ y 
+\begin_inset Formula $dt=\sqrt{-\frac{d}{a}}\sec u\tan u\,du$
+\end_inset
+
+.
+ También podemos hacer 
+\begin_inset Formula $at^{2}=-d\cosh^{2}u$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\sqrt{at^{2}+d}=[...]\sqrt{-d}\sinh u$
+\end_inset
+
+ y 
+\begin_inset Formula $dt=\sqrt{-\frac{d}{a}}\sinh u\,du$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\backslash
+sremember{FUVR2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Si 
+\begin_inset Formula $a<0$
+\end_inset
+
+ y 
+\begin_inset Formula $d>0$
+\end_inset
+
+ hacemos 
+\begin_inset Formula $at^{2}=-d\sin^{2}u$
+\end_inset
+
+ y entonces 
+\begin_inset Formula $\sqrt{at^{2}+d}=[..]\sqrt{d}\cos u$
+\end_inset
+
+ y 
+\begin_inset Formula $dt=\sqrt{-\frac{d}{a}}\cos u\,du$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+eremember
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Cambio de variable
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ un abierto y 
+\begin_inset Formula $T:\Omega\subseteq\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+, llamamos 
+\series bold
+jacobiano
+\series default
+ de 
+\begin_inset Formula $T$
+\end_inset
+
+ en 
+\begin_inset Formula $a\in\Omega$
+\end_inset
+
+ a la matriz cuadrada asociada a 
+\begin_inset Formula $dT(a)$
+\end_inset
+
+, 
+\begin_inset Formula $\left(\frac{\partial T_{i}}{\partial x_{j}}(a)\right)_{ij}$
+\end_inset
+
+.
+ El 
+\series bold
+teorema de cambio de variable
+\series default
+ afirma que si 
+\begin_inset Formula $\Omega\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ es un abierto medible Jordan y 
+\begin_inset Formula $T:\Omega\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ es una función inyectiva diferenciable con derivadas parciales continuas
+ tal que 
+\begin_inset Formula $\forall x\in\Omega,\det(dg(x))\neq0$
+\end_inset
+
+, si 
+\begin_inset Formula $f:T(\Omega)\rightarrow\mathbb{R}$
+\end_inset
+
+ es integrable Riemann en 
+\begin_inset Formula $T(\Omega)$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f\circ T$
+\end_inset
+
+ es integrable Riemann en 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\int_{g(A)}f(\vec{x})d\vec{x}=\int f(T(\vec{y}))|\det(dg(\vec{y}))|d\vec{y}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Algunos cambios de variable importantes:
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Coordenadas polares
+\series default
+ en 
+\begin_inset Formula $\mathbb{R}^{2}$
+\end_inset
+
+: Los puntos vienen dados por la distancia al origen, y el ángulo entre
+ el eje OX y el vector desde el origen al punto.
+ La función de cambio de variable es 
+\begin_inset Formula $T(\rho,\theta)=(\rho\cos\theta,\rho\sin\theta)$
+\end_inset
+
+, inyectiva en cualquier banda de la forma 
+\begin_inset Formula $(0,+\infty)\times(a,b)$
+\end_inset
+
+ con 
+\begin_inset Formula $b-a\leq2\pi$
+\end_inset
+
+, y
+\begin_inset Formula 
+\[
+|dT(\rho,\theta)|=\left|\begin{array}{cc}
+\cos\theta & -\rho\sin\theta\\
+\sin\theta & \rho\cos\theta
+\end{array}\right|=\rho
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Coordenadas cilíndricas
+\series default
+ en 
+\begin_inset Formula $\mathbb{R}^{3}$
+\end_inset
+
+: Los puntos vienen dados por las coordenadas de 
+\begin_inset Formula $(x,y)$
+\end_inset
+
+ en polares y la coordenada 
+\begin_inset Formula $z$
+\end_inset
+
+.
+ La función de cambio es 
+\begin_inset Formula $T(\rho,\theta,z)=(\rho\cos\theta,\rho\sin\theta,z)$
+\end_inset
+
+, inyectiva en cualquier banda de la forma 
+\begin_inset Formula $(0,+\infty)\times(a,b)\times\mathbb{R}$
+\end_inset
+
+ con 
+\begin_inset Formula $b-a\leq2\pi$
+\end_inset
+
+, y
+\begin_inset Formula 
+\[
+|dT(\rho,\theta,z)|=\left|\begin{array}{ccc}
+\cos\theta & -\rho\sin\theta & 0\\
+\sin\theta & \rho\cos\theta & 0\\
+0 & 0 & 1
+\end{array}\right|=\rho
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+
+\series bold
+Coordenadas esféricas
+\series default
+ o 
+\series bold
+polares
+\series default
+ en 
+\begin_inset Formula $\mathbb{R}^{3}$
+\end_inset
+
+: Los puntos vienen dados por la distancia al origen; el ángulo entre el
+ eje OX y la proyección en el plano XY del vector del origen al punto, y
+ el ángulo entre el eje OZ y el vector del origen al punto.
+ La función de cambio es 
+\begin_inset Formula $T(\rho,\theta,\varphi)=(\rho\cos\theta\sin\varphi,\rho\sin\theta\sin\varphi,\rho,\rho\cos\varphi)$
+\end_inset
+
+, inyectiva en cualquier banda de la forma 
+\begin_inset Formula $(0,+\infty)\times(a,b)\times(0,\pi)$
+\end_inset
+
+ con 
+\begin_inset Formula $b-a\leq2\pi$
+\end_inset
+
+, y
+\begin_inset Formula 
+\[
+|dT(\rho,\theta,\varphi)|=\left|\begin{array}{ccc}
+\cos\theta\sin\varphi & -\rho\sin\theta\sin\varphi & \rho\cos\theta\cos\varphi\\
+\sin\theta\sin\varphi & \rho\cos\theta\sin\varphi & \rho\sin\theta\cos\varphi\\
+\cos\varphi & 0 & -\rho\sin\varphi
+\end{array}\right|=\rho^{2}\sin\varphi
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+La integral de Riemann-Stieltjes
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $f,\varphi:[a,b]\rightarrow\mathbb{R}$
+\end_inset
+
+ y 
+\begin_inset Formula $P=\{a=t_{0}<\dots<t_{n}=b\}$
+\end_inset
+
+ una partición del intervalo, llamamos 
+\series bold
+suma de Riemann-Stieltjes
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ con respecto a 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ asociada a la partición 
+\begin_inset Formula $P$
+\end_inset
+
+ a cualquier suma de la forma
+\begin_inset Formula 
+\[
+R(f,\varphi,P,\{\xi_{i}\})=\sum_{i=1}^{n}f(\xi_{i})(\varphi(t_{i})-\varphi(t_{i-1}))
+\]
+
+\end_inset
+
+donde 
+\begin_inset Formula $\xi_{i}\in[t_{i-1},t_{i}]\forall i\in\{1,\dots,n\}$
+\end_inset
+
+.
+ Si existe el límite de estas sumas cuando 
+\begin_inset Formula $|P|:=\sup\{t_{i}-t_{i-1}\}_{i=1}^{n}$
+\end_inset
+
+ tiende a 0 se dice que 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\series bold
+integrable en el sentido de Riemann-Stieltjes
+\series default
+ con respecto a 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ en el intervalo 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+, y a este límite lo llamamos 
+\series bold
+integral de Riemann-Stieltjes
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ con respecto a 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+, denotado como
+\begin_inset Formula 
+\[
+\int_{a}^{b}f(x)d\varphi(x)=\int_{a}^{b}f\,d\varphi
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vemos que si 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ es la identidad entonces la integral es exactamente la de Riemann.
+ Denotamos 
+\begin_inset Formula $\lambda_{\varphi}([a,b]):=\varphi(b)-\varphi(a)$
+\end_inset
+
+.
+ Para que esta medida sea positiva es necesario que 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ sea creciente.
+ Una función 
+\begin_inset Formula $\varphi:D\subseteq\mathbb{R}\rightarrow\mathbb{R}$
+\end_inset
+
+ es 
+\series bold
+continua por la derecha
+\series default
+ en 
+\begin_inset Formula $c\in D$
+\end_inset
+
+ si 
+\begin_inset Formula $\lim_{x\rightarrow c^{+}}\varphi(x)=\varphi(c)$
+\end_inset
+
+.
+ Si se quiere que 
+\begin_inset Formula $\lambda_{\varphi}$
+\end_inset
+
+ se comporte bien al hacer uniones crecientes o intersecciones decrecientes
+ de intervalos, es necesario que 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ sea continua por la derecha.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable Riemann-Stieltjes con respecto a 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $\forall\varepsilon>0,\exists\delta>0:\forall P,Q\in[a,b],(|P|,|Q|<\delta\implies|R(f,\varphi,P)-R(f,\varphi,Q)|<\varepsilon)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Dado 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, existe 
+\begin_inset Formula $\delta>0$
+\end_inset
+
+ tal que si 
+\begin_inset Formula $|P|<\delta$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\left|R(f,\varphi,P)-\int_{a}^{b}f\,d\varphi\right|<\frac{\varepsilon}{2}$
+\end_inset
+
+, pero si 
+\begin_inset Formula $P$
+\end_inset
+
+ y 
+\begin_inset Formula $Q$
+\end_inset
+
+ son son particiones de 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ con 
+\begin_inset Formula $|P|,|Q|<\delta$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+|R(f,\varphi,P)-R(f,\varphi,Q)|\leq\left|R(f,\varphi,P)-\int_{a}^{b}f\,d\varphi\right|+\left|\int_{a}^{b}f\,d\varphi-R(f,\varphi,Q)\right|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Dada una sucesión de particiones 
+\begin_inset Formula $(P_{k})_{k}$
+\end_inset
+
+ de 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ con 
+\begin_inset Formula $|P_{k}|\rightarrow0$
+\end_inset
+
+, por la condición se tiene que 
+\begin_inset Formula $(R(f,\varphi,P_{k}))_{k}$
+\end_inset
+
+ es de Cauchy y por tanto converge hacia un 
+\begin_inset Formula $I\in\mathbb{R}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ y 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ es monótona creciente definida en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable Riemann-Stieltjes respecto a 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Al ser 
+\begin_inset Formula $f$
+\end_inset
+
+ continua en un compacto, es uniformemente continua, luego para 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+ existe 
+\begin_inset Formula $\delta>0$
+\end_inset
+
+ tal que si 
+\begin_inset Formula $|x-y|<\delta$
+\end_inset
+
+ entonces 
+\begin_inset Formula $|f(x)-f(y)|<\frac{\varepsilon}{\varphi(b)-\varphi(a)}$
+\end_inset
+
+.
+ Sean 
+\begin_inset Formula $P:=\{a=x_{0}<\dots<x_{n}=b\}$
+\end_inset
+
+ y 
+\begin_inset Formula $Q:=\{a=y_{0}<\dots<y_{m}=b\}$
+\end_inset
+
+ particiones con 
+\begin_inset Formula $|P|,|Q|<\frac{\delta}{2}$
+\end_inset
+
+ y 
+\begin_inset Formula $P\lor Q=:\{a=t_{0}<\dots<t_{p}=b\}$
+\end_inset
+
+, podemos escribir las sumas de Riemann-Stieltjes asociadas a 
+\begin_inset Formula $P$
+\end_inset
+
+ y 
+\begin_inset Formula $Q$
+\end_inset
+
+ como 
+\begin_inset Formula $R(f,\varphi,P)=\sum_{i=1}^{n}f(\chi_{i})(\varphi(x_{i})-\varphi(x_{i-1}))=\sum_{j=1}^{p}f(\chi_{j}^{*})(\varphi(t_{j})-\varphi(t_{j-1}))$
+\end_inset
+
+ y 
+\begin_inset Formula $R(f,\varphi,Q)=\sum_{k=1}^{m}f(\eta_{i})(\varphi(y_{k})-\varphi(y_{k-1}))=\sum_{j=1}^{p}f(\eta_{j}^{*})(\varphi(t_{j})-\varphi(t_{j-1}))$
+\end_inset
+
+, respectivamente, donde 
+\begin_inset Formula $\chi_{j}^{*}=\chi_{i}$
+\end_inset
+
+ si 
+\begin_inset Formula $[t_{j-1},t_{j}]\subseteq[x_{i-1},x_{i}]$
+\end_inset
+
+ y 
+\begin_inset Formula $\eta_{j}^{*}=\eta_{k}$
+\end_inset
+
+ si 
+\begin_inset Formula $[t_{j-1},t_{j}]\subseteq[y_{k-1},y_{k}]$
+\end_inset
+
+.
+ De aquí, 
+\begin_inset Formula $|\chi_{j}^{*}-\eta_{j}^{*}|\leq|\chi_{j}^{*}-t_{j}|+|t_{j}-\eta_{j}^{*}|<\frac{\delta}{2}+\frac{\delta}{2}=\delta$
+\end_inset
+
+, y con esto 
+\begin_inset Formula 
+\begin{multline*}
+|R(f,\varphi,P)-R(f,\varphi,Q)|\leq\sum_{j=1}^{p}|f(\chi_{j}^{*})-f(\eta_{j}^{*})|(\varphi(t_{j})-\varphi(t_{j-1}))<\\
+<\sum_{j=1}^{p}\frac{\varepsilon}{\varphi(b)-\varphi(a)}(\varphi(t_{j})-\varphi(t_{j-1}))=\frac{\varepsilon}{\varphi(b)-\varphi(a)}(\varphi(b)-\varphi(a))=\varepsilon
+\end{multline*}
+
+\end_inset
+
+Por tanto si 
+\begin_inset Formula $f$
+\end_inset
+
+ es continua en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ y 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ es una función creciente de clase 
+\begin_inset Formula ${\cal C}^{1}$
+\end_inset
+
+ en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+, se cumple que
+\begin_inset Formula 
+\[
+\int_{a}^{b}f(x)d\varphi(x)=\int_{a}^{b}f(x)\varphi'(x)dx
+\]
+
+\end_inset
+
+ Propiedades: Dadas 
+\begin_inset Formula $f,g:[a,b]\rightarrow\mathbb{R}$
+\end_inset
+
+ integrables Riemann-Stieltjes en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ respecto a 
+\begin_inset Formula $\varphi,\psi:[a,b]\rightarrow\mathbb{R}$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda\in\mathbb{R}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\int_{a}^{b}(\lambda f)d\varphi=\int_{a}^{b}f\,d(\lambda\varphi)=\lambda\int_{a}^{b}f\,d\varphi$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\int_{a}^{b}(f+g)d\varphi=\int_{a}^{b}f\,d\varphi+\int_{a}^{b}g\,d\varphi$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f(t)\geq g(t)\forall t\in[a,b]$
+\end_inset
+
+ y 
+\begin_inset Formula $\varphi$
+\end_inset
+
+ es monótona creciente, 
+\begin_inset Formula $\int_{a}^{b}f\,d\varphi\geq\int_{a}^{b}g\,d\varphi$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\int_{a}^{b}f\,d(\varphi+\psi)=\int_{a}^{b}f\,d\varphi+\int_{a}^{b}f\,d\psi$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $c\in(a,b)$
+\end_inset
+
+, 
+\begin_inset Formula $\int_{a}^{c}f\,d\varphi+\int_{c}^{b}f\,d\varphi=\int_{a}^{b}f\,d\varphi$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+integración por partes
+\series default
+ en Riemann-Stieltjes se basa en que, si existe 
+\begin_inset Formula $\int_{a}^{b}f\,d\varphi$
+\end_inset
+
+ entonces también existe 
+\begin_inset Formula $\int_{a}^{b}\varphi\,df$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\int_{a}^{b}\varphi\,df=(f(b)\varphi(b)-f(a)\varphi(a))-\int_{a}^{b}f\,d\varphi
+\]
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $P=\{a=t_{0}<\dots<t_{n}=b\}$
+\end_inset
+
+ una partición de 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ y 
+\begin_inset Formula $\xi_{i}\in[t_{i-1},t_{i}]\forall i\in\{1,\dots,n\}$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $Q=\{a=\xi_{0}<\dots<\xi_{n}<\xi_{n+1}=b\}$
+\end_inset
+
+, 
+\begin_inset Formula $x_{1}:=t_{0}=a\in[\xi_{0},\xi_{1}]$
+\end_inset
+
+, 
+\begin_inset Formula $x_{i}:=t_{i-1}\in[\xi_{i-1},\xi_{i}]\forall i\in\{1,\dots,n\}$
+\end_inset
+
+ y 
+\begin_inset Formula $x_{n+1}:=t_{n}=b\in[\xi_{n},\xi_{n+1}]$
+\end_inset
+
+.
+ Entonces
+\begin_inset Formula 
+\begin{multline*}
+R(\varphi,f,P,\{\xi_{i}\})=\sum_{i=1}^{n}\varphi(\xi_{i})(f(t_{i})-f(t_{i-1}))=\sum_{i=1}^{n}f(t_{i})\varphi(\xi_{i})-\sum_{i=1}^{n}f(t_{i-1})\varphi(\xi_{i})=\\
+=\sum_{i=1}^{n-1}f(t_{i})\varphi(\xi_{i})+f(b)\varphi(b)-f(a)\varphi(\xi_{1})-\sum_{i=2}^{n}f(t_{i-1})\varphi(\xi_{i})=\\
+=f(b)\varphi(b)-f(a)\varphi(a)-f(a)(\varphi(\xi_{1})-\varphi(a))+\sum_{i=2}^{n}f(x_{i})\varphi(\xi_{i-1})-\sum_{i=2}^{n}f(x_{i})\varphi(\xi_{i})=\\
+=f(b)\varphi(b)-f(a)\varphi(a)-\sum_{i=1}^{n}f(x_{i})(\varphi(\xi_{i})-\varphi(\xi_{i-1}))=f(b)\varphi(b)-f(a)\varphi(a)-R(f,\varphi,Q,\{x_{i}\})
+\end{multline*}
+
+\end_inset
+
+Basta pues tomar límites en esta última expresión cuando 
+\begin_inset Formula $|Q|\leq2|P|\rightarrow0$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
diff --git a/fvv2/n3.lyx b/fvv2/n3.lyx
new file mode 100644
index 0000000..a35f67f
--- /dev/null
+++ b/fvv2/n3.lyx
@@ -0,0 +1,3763 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 0
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Section
+Funciones medibles
+\end_layout
+
+\begin_layout Standard
+Una aplicación 
+\begin_inset Formula $f:\Omega\rightarrow\Omega'$
+\end_inset
+
+ es 
+\series bold
+
+\begin_inset Formula $(\Sigma,\Sigma')$
+\end_inset
+
+-medible
+\series default
+ si 
+\begin_inset Formula $\forall E'\in\Sigma',f^{-1}(E')\in\Sigma$
+\end_inset
+
+.
+ Cuando 
+\begin_inset Formula $\Omega'$
+\end_inset
+
+ es un espacio topológico, decimos que 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\series bold
+
+\begin_inset Formula $\Sigma$
+\end_inset
+
+-medible
+\series default
+ si es 
+\begin_inset Formula $(\Sigma,{\cal B}(\Omega'))$
+\end_inset
+
+-medible.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\Sigma'=:\sigma({\cal D})$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\begin_inset Formula $(\Sigma,\Sigma')$
+\end_inset
+
+-medible si y sólo si 
+\begin_inset Formula $\forall D\in{\cal D},f^{-1}(D)\in\Sigma$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Obvio.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula ${\cal A}:=\{E\in\Sigma':f^{-1}(E)\in\Sigma\}$
+\end_inset
+
+, vemos que 
+\begin_inset Formula ${\cal A}$
+\end_inset
+
+ es una 
+\begin_inset Formula $\sigma$
+\end_inset
+
+-álgebra que contiene a 
+\begin_inset Formula ${\cal D}$
+\end_inset
+
+, pues 
+\begin_inset Formula $f^{-1}(E^{\complement})=\Omega\backslash f^{-1}(E)$
+\end_inset
+
+ y 
+\begin_inset Formula $f^{-1}(\bigcup_{k=1}^{\infty}E_{k})=\bigcup_{k=1}^{\infty}f^{-1}(E_{k})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sean 
+\begin_inset Formula $T:=(X,{\cal T})$
+\end_inset
+
+ y 
+\begin_inset Formula $T':=(X,{\cal T}')$
+\end_inset
+
+ espacios topológicos, toda función 
+\begin_inset Formula $f:T\rightarrow T'$
+\end_inset
+
+ continua es 
+\begin_inset Formula $({\cal B}(T),{\cal B}(T'))$
+\end_inset
+
+-medible, pues 
+\begin_inset Formula ${\cal B}(T')=\sigma({\cal T}')$
+\end_inset
+
+ y la continuidad asegura que 
+\begin_inset Formula $\forall A\in{\cal T}',f^{-1}(A)\in{\cal T}\subseteq{\cal B}(T)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Composición de medibles:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f:\Omega\rightarrow\Omega'$
+\end_inset
+
+ y 
+\begin_inset Formula $g:f(\Omega)\rightarrow\Omega''$
+\end_inset
+
+ son medibles, 
+\begin_inset Formula $g\circ f$
+\end_inset
+
+ también lo es.
+ En particular, si 
+\begin_inset Formula $T$
+\end_inset
+
+ y 
+\begin_inset Formula $T'$
+\end_inset
+
+ son espacios topológicos, 
+\begin_inset Formula $f:\Omega\rightarrow T$
+\end_inset
+
+ es 
+\begin_inset Formula $(\Sigma,{\cal B}(T))$
+\end_inset
+
+-medible y 
+\begin_inset Formula $g:T\rightarrow T'$
+\end_inset
+
+ es continua, 
+\begin_inset Formula $g\circ f$
+\end_inset
+
+ es 
+\begin_inset Formula $(\Sigma,{\cal B}(T'))$
+\end_inset
+
+-medible.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Para 
+\begin_inset Formula $E\in\Sigma''$
+\end_inset
+
+, 
+\begin_inset Formula $g^{-1}(E)\in\Sigma'$
+\end_inset
+
+ por ser 
+\begin_inset Formula $g$
+\end_inset
+
+ medible y 
+\begin_inset Formula $f^{-1}(g^{-1}(E))=(g\circ f)^{-1}(E)\in\Sigma$
+\end_inset
+
+ por ser 
+\begin_inset Formula $g$
+\end_inset
+
+ medible, luego 
+\begin_inset Formula $g\circ f$
+\end_inset
+
+ es medible.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $u_{1},\dots u_{n}:\Omega\rightarrow\mathbb{R}$
+\end_inset
+
+ son medibles, 
+\begin_inset Formula $\varphi:\Omega\rightarrow\mathbb{R}^{n}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\varphi(\omega):=(u_{1}(\omega),\dots,u_{n}(\omega))$
+\end_inset
+
+ es medible.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Un abierto 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ se puede expresar como unión numerable de cubos diádicos disjuntos, y como
+ estos están en 
+\begin_inset Formula ${\cal B}(\mathbb{R}^{n})$
+\end_inset
+
+, lo generan.
+ Si 
+\begin_inset Formula $[a,b)$
+\end_inset
+
+ es uno de estos cubos,
+\begin_inset Formula 
+\[
+\varphi^{-1}([a,b))=\bigcap_{i=1}^{n}u_{i}^{-1}([a_{i},b_{i}))\in\Sigma
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $f:\Omega\rightarrow\mathbb{C}$
+\end_inset
+
+ es medible si y sólo si lo son 
+\begin_inset Formula $\omega\mapsto\text{Re}(f(\omega))$
+\end_inset
+
+ e 
+\begin_inset Formula $\omega\mapsto\text{Im}(f(\omega))$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+ Se deriva de que 
+\begin_inset Formula $x\mapsto\text{Re}(x)$
+\end_inset
+
+ y 
+\begin_inset Formula $x\mapsto\text{Im}(x)$
+\end_inset
+
+ son continuas.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{sloppypar}
+\end_layout
+
+\end_inset
+
+Sea 
+\begin_inset Formula $\pi:\mathbb{R}^{2}\rightarrow\mathbb{C}$
+\end_inset
+
+ el isomorfismo canónico, como 
+\begin_inset Formula $\pi$
+\end_inset
+
+ es continua y 
+\begin_inset Formula $f(\omega)=\pi(\text{Re}(f(\omega)),\text{Im}(f(\omega)))$
+\end_inset
+
+, el punto anterior nos da que 
+\begin_inset Formula $f$
+\end_inset
+
+ es medible.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{sloppypar}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f,g:\Omega\rightarrow\mathbb{C}$
+\end_inset
+
+ son medibles, 
+\begin_inset Formula $f+g$
+\end_inset
+
+, 
+\begin_inset Formula $fg$
+\end_inset
+
+ y 
+\begin_inset Formula $\frac{f}{g}\chi_{\{g\neq0\}}$
+\end_inset
+
+ también lo son.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Se debe a que 
+\begin_inset Formula $(x,y)\mapsto x+y$
+\end_inset
+
+, 
+\begin_inset Formula $(x,y)\mapsto xy$
+\end_inset
+
+ son continuas.
+ Para el cociente, si 
+\begin_inset Formula $\{g=0\}=\emptyset$
+\end_inset
+
+, la continuidad de 
+\begin_inset Formula $(x,y)\mapsto\frac{x}{y}$
+\end_inset
+
+ cuando 
+\begin_inset Formula $y\neq0$
+\end_inset
+
+ implica que 
+\begin_inset Formula $\frac{f}{g}$
+\end_inset
+
+ es medible.
+ En el caso general, sea 
+\begin_inset Formula $S\subseteq\mathbb{C}$
+\end_inset
+
+ medible,
+\begin_inset Formula 
+\begin{multline*}
+\left(\frac{f}{g}\chi_{\{g\neq0\}}\right)^{-1}(S)=\left(\left(\frac{f}{g}\chi_{\{g\neq0\}}\right)^{-1}(S)\cap\{g\neq0\}\right)\cup\left(\left(\frac{f}{g}\chi_{\{g\neq0\}}\right)^{-1}(S)\cap\{g=0\}\right)\\
+=\left(\frac{f}{g}\chi_{\{g\neq0\}}\right)^{-1}(S\backslash\{0\}))\cup\left(\frac{f}{g}\chi_{\{g\neq0\}}\right)^{-1}(S\cap\{0\})
+\end{multline*}
+
+\end_inset
+
+El primer elemento de esta última unión no contiene elementos de 
+\begin_inset Formula $\{g=0\}$
+\end_inset
+
+, pues para todos ellos 
+\begin_inset Formula $\frac{f}{g}\chi_{\{g\neq0\}}(\omega)\neq0$
+\end_inset
+
+, por tanto el conjunto es igual a 
+\begin_inset Formula $\left(\frac{f}{g}\right)^{-1}(S\backslash\{0\})$
+\end_inset
+
+, que es medible.
+ El segundo elemento es 
+\begin_inset Formula $\emptyset$
+\end_inset
+
+ si 
+\begin_inset Formula $0\notin S$
+\end_inset
+
+ o 
+\begin_inset Formula $\{f=0\}\cup\{g=0\}$
+\end_inset
+
+ si 
+\begin_inset Formula $0\in S$
+\end_inset
+
+; en cualquier caso es medible.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f:\Omega\rightarrow\mathbb{C}$
+\end_inset
+
+ es medible, 
+\begin_inset Formula $|f|$
+\end_inset
+
+ también lo es y existe 
+\begin_inset Formula $\alpha:\Omega\rightarrow\mathbb{C}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\forall\omega\in\Omega,|\alpha(\omega)|=1$
+\end_inset
+
+ y 
+\begin_inset Formula $|f|=\alpha f$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $|f|$
+\end_inset
+
+ es medible por ser 
+\begin_inset Formula $\omega\mapsto\sqrt{\text{Re}(f(\omega))^{2}+\text{Im}(f(\omega))^{2}}$
+\end_inset
+
+, al igual que 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\alpha(\omega)=\chi_{\{f=0\}}+\frac{f}{|f|}\chi_{\{f\neq0\}}$
+\end_inset
+
+, que cumple las condiciones.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Una función 
+\begin_inset Formula $f:\Omega\rightarrow[-\infty,+\infty]$
+\end_inset
+
+ es 
+\series bold
+
+\begin_inset Formula $\Sigma$
+\end_inset
+
+-medible
+\series default
+ si es 
+\begin_inset Formula $(\Sigma,\Sigma':=\sigma(\{(a,+\infty]\}_{a\in\mathbb{R}}))$
+\end_inset
+
+-medible.
+ Todos los cubos diádicos 
+\begin_inset Formula $[a,b)\subseteq\mathbb{R}$
+\end_inset
+
+ están en 
+\begin_inset Formula $\Sigma'$
+\end_inset
+
+, luego también están todos los abiertos de 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ como unión numerable de cubos diádicos y por tanto 
+\begin_inset Formula ${\cal B}(\mathbb{R})\subseteq\Sigma'$
+\end_inset
+
+.
+ Adoptamos el convenio 
+\begin_inset Formula $\pm\infty\cdot0=0\cdot\pm\infty=0$
+\end_inset
+
+ y la notación 
+\begin_inset Formula $\{f\bullet a\}:=\{\omega\in\Omega:f(\omega)\bullet a\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $f:\Omega\rightarrow[-\infty,+\infty]$
+\end_inset
+
+, si tenemos que
+\begin_inset Formula 
+\begin{multline*}
+f\text{ es medible}\iff\forall a\in\mathbb{R},\{f>a\}\in\Sigma\iff\forall a\in\mathbb{R},\{f\geq a\}\in\Sigma\\
+\iff\forall a\in\mathbb{R},\{f<a\}\in\Sigma\iff\forall a\in\mathbb{R},\{f\leq a\}\in\Sigma
+\end{multline*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $1\implies3]$
+\end_inset
+
+ Como 
+\begin_inset Formula $(a,+\infty]$
+\end_inset
+
+ es medible, 
+\begin_inset Formula $f^{-1}((a,+\infty])=\{f>a\}$
+\end_inset
+
+ también lo es.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $3\implies1]$
+\end_inset
+
+ Para cada elemento de 
+\begin_inset Formula $\{(a,+\infty]\}_{a\in\mathbb{R}}$
+\end_inset
+
+, 
+\begin_inset Formula $f^{-1}((a,+\infty])=\{f>a\}\in\Sigma$
+\end_inset
+
+, y por tanto 
+\begin_inset Formula $f$
+\end_inset
+
+ es medible.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $2\implies3]$
+\end_inset
+
+ 
+\begin_inset Formula $\{f\geq a\}=\bigcap_{k=1}^{\infty}\{f>a-\frac{1}{k}\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $3\implies4]$
+\end_inset
+
+ 
+\begin_inset Formula $\{f<a\}=\Omega\backslash\{f\geq a\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $4\implies5]$
+\end_inset
+
+ 
+\begin_inset Formula $\{f\leq a\}=\bigcap_{k=1}^{\infty}\{f<a+\frac{1}{k}\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $5\implies2]$
+\end_inset
+
+ 
+\begin_inset Formula $\{f>a\}=\Omega\backslash\{f\leq a\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Además, si 
+\begin_inset Formula $f$
+\end_inset
+
+ es medible, 
+\begin_inset Formula $\{f>-\infty\}$
+\end_inset
+
+, 
+\begin_inset Formula $\{f<+\infty\}$
+\end_inset
+
+, 
+\begin_inset Formula $\{f=-\infty\}$
+\end_inset
+
+, 
+\begin_inset Formula $\{f=+\infty\}$
+\end_inset
+
+, 
+\begin_inset Formula $\{a\leq f\leq b\}$
+\end_inset
+
+, 
+\begin_inset Formula $\{f=a\}$
+\end_inset
+
+, etc.
+ son medibles, y 
+\begin_inset Formula $f$
+\end_inset
+
+ es medible si y sólo si 
+\begin_inset Formula $\{f=-\infty\}$
+\end_inset
+
+ es medible y para 
+\begin_inset Formula $a\in\mathbb{R}$
+\end_inset
+
+,
+\begin_inset Formula $\{a<f<+\infty\}$
+\end_inset
+
+ es medible.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\{a<f<+\infty\}=\{f>a\}\cap\bigcup_{k=1}^{\infty}\{f<k\}$
+\end_inset
+
+, 
+\begin_inset Formula $\{f=-\infty\}=\bigcap_{k=1}^{\infty}\{f<-k\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Formula $\{f>a\}=\{a<f<+\infty\}\cup\left(\{f=-\infty\}\cup\bigcup_{k=1}^{\infty}\{-k<f<+\infty\}\right)^{\complement}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dado un subconjunto denso 
+\begin_inset Formula $A\subseteq[-\infty,+\infty]$
+\end_inset
+
+, si 
+\begin_inset Formula $\{f>a\}$
+\end_inset
+
+ es medible para todo 
+\begin_inset Formula $a\in A$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es medible.
+ 
+\series bold
+Demostración:
+\series default
+ Para 
+\begin_inset Formula $a\in\mathbb{R}$
+\end_inset
+
+, existe una sucesión 
+\begin_inset Formula $(a_{k})_{k}$
+\end_inset
+
+ de elementos de 
+\begin_inset Formula $A$
+\end_inset
+
+ que converge a 
+\begin_inset Formula $a$
+\end_inset
+
+ con 
+\begin_inset Formula $a_{k}>a\forall k\in\mathbb{N}$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $\{f>a\}=\bigcup\{f>a_{k}\}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f:\Omega\rightarrow[-\infty,+\infty]$
+\end_inset
+
+ es medible si y sólo si para todo abierto 
+\begin_inset Formula $G\subseteq\mathbb{R}^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $f^{-1}(G)$
+\end_inset
+
+ es medible.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Los abiertos de 
+\begin_inset Formula $\mathbb{R}^{n}$
+\end_inset
+
+ están en 
+\begin_inset Formula $\sigma(\{(a,+\infty]\}_{a\in\mathbb{R}})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Para todo 
+\begin_inset Formula $a\in\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $f^{-1}((a,+\infty))=\{a<f<+\infty\}$
+\end_inset
+
+ es medible, luego 
+\begin_inset Formula $f$
+\end_inset
+
+ lo es.
+\end_layout
+
+\begin_layout Standard
+Una propiedad se cumple 
+\series bold
+en casi todo punto
+\series default
+ de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ si el conjunto de puntos 
+\begin_inset Formula $\omega\in\Omega$
+\end_inset
+
+ en los que no se cumple tiene medida cero.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f,g:\Omega\rightarrow[-\infty,+\infty]$
+\end_inset
+
+ son medibles, 
+\begin_inset Formula $\{f>g\}$
+\end_inset
+
+ también lo es.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si 
+\begin_inset Formula $\{r_{k}\}_{k\in\mathbb{N}}=\mathbb{Q}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\{f>g\}=\bigcup_{k=1}^{\infty}\{f>r_{k}>g\}=\bigcup_{k=1}^{\infty}(\{f>r_{k}\}\cap\{g<r_{k}\})$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es medible y 
+\begin_inset Formula $g=f$
+\end_inset
+
+ en casi todo punto, 
+\begin_inset Formula $g$
+\end_inset
+
+ es medible y 
+\begin_inset Formula $\mu(\{g>a\})=\mu(\{f>a\})$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si 
+\begin_inset Formula $Z:=\{f\neq g\}$
+\end_inset
+
+, 
+\begin_inset Formula $\{g>a\}\cup Z=\{f>a\}\cup Z$
+\end_inset
+
+, luego si 
+\begin_inset Formula $f$
+\end_inset
+
+ es medible, 
+\begin_inset Formula $\{g>a\}\cup Z$
+\end_inset
+
+ lo es y como difiere de 
+\begin_inset Formula $\{g>a\}$
+\end_inset
+
+ por un conjunto de medida cero, 
+\begin_inset Formula $g$
+\end_inset
+
+ es medible.
+ Por último, 
+\begin_inset Formula $\mu(\{g>a\})=\mu(\{g>a\}\cup Z)=\mu(\{f>a\}\cup Z)=\mu(\{f>a\})$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es medible y 
+\begin_inset Formula $\lambda\in\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $f+\lambda$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda f$
+\end_inset
+
+ son medibles.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Para 
+\begin_inset Formula $a\in\mathbb{R}$
+\end_inset
+
+, 
+\begin_inset Formula $\{f+\lambda>a\}=\{f>a-\lambda\}$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\{\lambda f>a\}=\begin{cases}
+f>\frac{a}{\lambda} & \text{si }\lambda>0\\
+f<\frac{a}{\lambda} & \text{si }\lambda<0\\
+\Omega & \text{si }\lambda=0\land a<0\\
+\emptyset & \text{si }\lambda=0\land a\geq0
+\end{cases}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ y 
+\begin_inset Formula $g$
+\end_inset
+
+ son medibles, también lo es 
+\begin_inset Formula $f+g$
+\end_inset
+
+.
+ Esto significa que las combinaciones lineales 
+\begin_inset Formula $\lambda_{1}f_{1}+\dots+\lambda_{n}f_{n}$
+\end_inset
+
+ de funciones medibles 
+\begin_inset Formula $f_{1},\dots,f_{n}$
+\end_inset
+
+ son medibles, por lo que el conjunto de funciones 
+\begin_inset Formula $\Omega\rightarrow[-\infty,+\infty]$
+\end_inset
+
+ 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+-medibles es un espacio vectorial.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Para 
+\begin_inset Formula $a\in\mathbb{R}$
+\end_inset
+
+, si 
+\begin_inset Formula $g$
+\end_inset
+
+ es medible, también lo es 
+\begin_inset Formula $a-g$
+\end_inset
+
+, y entonces 
+\begin_inset Formula $\{f+g>a\}=\{f>a-g\}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Para 
+\begin_inset Formula $f,g:\Omega\rightarrow[-\infty,+\infty]$
+\end_inset
+
+ medibles, 
+\begin_inset Formula $fg$
+\end_inset
+
+ es medible y, si 
+\begin_inset Formula $g\neq0$
+\end_inset
+
+ en casi todo punto, 
+\begin_inset Formula $f/g$
+\end_inset
+
+ (definida en casi todo punto) es medible.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es medible, es fácil ver que 
+\begin_inset Formula $f^{2}=|f|^{2}$
+\end_inset
+
+ también lo es.
+ Para valores finitos, 
+\begin_inset Formula $fg=\frac{(f+g)^{2}-(f-g)^{2}}{4}$
+\end_inset
+
+.
+ Si alguna de las funciones puede tener valores infinitos, vemos que 
+\begin_inset Formula $\{f=+\infty\}$
+\end_inset
+
+, 
+\begin_inset Formula $\{f=-\infty\}$
+\end_inset
+
+, 
+\begin_inset Formula $\{f=0\}$
+\end_inset
+
+, 
+\begin_inset Formula $\{g=+\infty\}$
+\end_inset
+
+, etc.
+ son medibles y por tanto 
+\begin_inset Formula $\{fg=0\}=\{f=0\}\cup\{g=0\}$
+\end_inset
+
+, 
+\begin_inset Formula $\{fg=+\infty\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\{fg=-\infty\}$
+\end_inset
+
+ también lo son.
+ Para 
+\begin_inset Formula $f/g$
+\end_inset
+
+, basta ver que 
+\begin_inset Formula $1/g$
+\end_inset
+
+ es medible, pero dado 
+\begin_inset Formula $a\in\mathbb{R}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\{1/g>a\}=\begin{cases}
+\{0<g<\frac{1}{a}\} & \text{si }a>0\\
+\{0<g<+\infty\} & \text{si }a=0\\
+\{g<\frac{1}{a}\}\cup\{0<g<+\infty\} & \text{si }a<0
+\end{cases}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Dada una sucesión de funciones medibles 
+\begin_inset Formula $(f_{k}:\Omega\rightarrow\mathbb{R})_{k}$
+\end_inset
+
+, las funciones 
+\begin_inset Formula $\sup_{k}f_{k}(\omega)$
+\end_inset
+
+, 
+\begin_inset Formula $\inf_{k}f_{k}(\omega)$
+\end_inset
+
+, 
+\begin_inset Formula $\limsup_{k}f_{k}(\omega)$
+\end_inset
+
+ y 
+\begin_inset Formula $\liminf_{k}f_{k}(\omega)$
+\end_inset
+
+ son medibles.
+ De aquí que, si 
+\begin_inset Formula $E$
+\end_inset
+
+ es el conjunto de puntos 
+\begin_inset Formula $\omega\in\Omega$
+\end_inset
+
+ en los que 
+\begin_inset Formula $(f_{n}(\omega))_{n}$
+\end_inset
+
+ converge, entonces el límite puntual de 
+\begin_inset Formula $(f_{n}|_{E})_{n}$
+\end_inset
+
+ es medible y 
+\begin_inset Formula $E$
+\end_inset
+
+ también lo es.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{sloppypar}
+\end_layout
+
+\end_inset
+
+Basta probarlo para el supremo, pues 
+\begin_inset Formula $\inf_{k}f_{k}(x)=-\sup_{k}(-f_{k})$
+\end_inset
+
+, 
+\begin_inset Formula $\limsup_{n}f_{n}(\omega)=\inf_{m\in\mathbb{N}}\{\sup_{k\geq m}\{f_{k}(\omega)\}\}$
+\end_inset
+
+ y 
+\begin_inset Formula $\liminf_{n}f_{n}(\omega)=\sup_{m\in\mathbb{N}}\{\inf_{k\geq m}\{f_{k}(\omega)\}\}$
+\end_inset
+
+, pero 
+\begin_inset Formula $\{\sup_{k}f_{k}>a\}=\bigcup_{k}\{f_{k}>a\}$
+\end_inset
+
+.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{sloppypar}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Section
+Funciones simples
+\end_layout
+
+\begin_layout Standard
+Una 
+\series bold
+función simple
+\series default
+ es aquella cuya imagen es finita.
+ Si 
+\begin_inset Formula $f:\Omega\rightarrow\Omega'$
+\end_inset
+
+ es una función simple que toma valores distintos 
+\begin_inset Formula $a_{1},\dots,a_{n}$
+\end_inset
+
+ en conjuntos disjuntos respectivos 
+\begin_inset Formula $E_{1},\dots,E_{n}$
+\end_inset
+
+, su 
+\series bold
+forma canónica
+\series default
+ es
+\begin_inset Formula 
+\[
+f=\sum_{k=1}^{n}a_{k}\chi_{E_{k}}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Claramente 
+\begin_inset Formula $\chi_{E}$
+\end_inset
+
+ es medible si y sólo si 
+\begin_inset Formula $E$
+\end_inset
+
+ lo es, por lo que 
+\begin_inset Formula $f$
+\end_inset
+
+ es medible si y sólo si lo son 
+\begin_inset Formula $E_{1},\dots,E_{n}$
+\end_inset
+
+.
+ Algunos autores sólo llaman funciones simples a las que además son medibles.
+ Como 
+\series bold
+teorema
+\series default
+:
+\end_layout
+
+\begin_layout Enumerate
+Toda función 
+\begin_inset Formula $f:\Omega\rightarrow[0,+\infty]$
+\end_inset
+
+ es límite de una sucesión creciente de funciones simples.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Para 
+\begin_inset Formula $k\in\mathbb{N}$
+\end_inset
+
+, subdividimos 
+\begin_inset Formula $[0,k]$
+\end_inset
+
+ en subintervalos 
+\begin_inset Formula $\{[(j-1)2^{-k},j2^{-k}]\}_{j\in\{1,\dots,k2^{k}\}}$
+\end_inset
+
+ y definimos
+\begin_inset Formula 
+\[
+f_{k}(x):=\begin{cases}
+\frac{j-1}{2^{k}} & \text{si }\frac{j-1}{2^{k}}\leq f(x)<\frac{j}{2^{k}},j\in\{1,\dots,k2^{k}\}\\
+k & \text{si }f(x)\geq k
+\end{cases}
+\]
+
+\end_inset
+
+Cada 
+\begin_inset Formula $f_{k}$
+\end_inset
+
+ es una función simple definida en todo 
+\begin_inset Formula $\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $f_{k}\leq f_{k+1}$
+\end_inset
+
+ porque al pasar de 
+\begin_inset Formula $f_{k}$
+\end_inset
+
+ a 
+\begin_inset Formula $f_{k+1}$
+\end_inset
+
+ dividimos los intervalos por la mitad y permitimos valores mayores en la
+ imagen, y 
+\begin_inset Formula $f_{k}\rightarrow f$
+\end_inset
+
+ porque para un valor 
+\begin_inset Formula $k$
+\end_inset
+
+ lo suficientemente grande, 
+\begin_inset Formula $0\leq f(x)-f_{k}(x)\leq2^{-k}$
+\end_inset
+
+ si 
+\begin_inset Formula $f(x)$
+\end_inset
+
+ es finito y 
+\begin_inset Formula $f_{k}=k\rightarrow+\infty$
+\end_inset
+
+ si 
+\begin_inset Formula $f(x)=+\infty$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Toda función 
+\begin_inset Formula $f:\Omega\rightarrow[-\infty,+\infty]$
+\end_inset
+
+ es límite de una sucesión de funciones simples.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Basta aplicar lo anterior a 
+\begin_inset Formula $f^{+}:=\max\{0,f\}$
+\end_inset
+
+ y 
+\begin_inset Formula $f^{-}:=-\min\{0,f\}$
+\end_inset
+
+ y restar las sucesiones resultantes.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si la función 
+\begin_inset Formula $f$
+\end_inset
+
+ en los dos apartados anteriores es medible, podemos hacer que las 
+\begin_inset Formula $f_{k}$
+\end_inset
+
+ también lo sean.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+En el primer apartado,
+\begin_inset Formula 
+\[
+f_{k}=\sum_{j=1}^{k2^{k}}\frac{j-1}{2^{k}}\chi_{\left\{ \frac{j-1}{2^{k}}\leq f<\frac{j}{2^{k}}\right\} }+k\chi_{\{f\geq k\}}
+\]
+
+\end_inset
+
+que es medible porque todos los conjuntos involucrados lo son, y en el segundo
+ basta considerar 
+\begin_inset Formula $f^{+}$
+\end_inset
+
+ y 
+\begin_inset Formula $f^{-}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es acotada, las funciones simples dadas convergerán uniformemente a 
+\begin_inset Formula $f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Integrales en funciones positivas
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $(\Omega,\Sigma,\mu)$
+\end_inset
+
+ un espacio de medida, 
+\begin_inset Formula ${\cal S}(\Omega)$
+\end_inset
+
+ el espacio vectorial de las funciones simples 
+\begin_inset Formula $\Omega\rightarrow[-\infty,+\infty]$
+\end_inset
+
+ y 
+\begin_inset Formula ${\cal S}(\Omega)^{+}:=\{h\in{\cal S}(\Omega):h\geq0\}$
+\end_inset
+
+, llamamos 
+\series bold
+integral
+\series default
+ de una función simple 
+\begin_inset Formula $h:=\sum_{k=1}^{n}a_{k}\chi_{E_{k}}\in{\cal S}(\Omega)^{+}$
+\end_inset
+
+ como
+\begin_inset Formula 
+\[
+\int h\,d\mu:=\sum_{k=1}^{n}a_{k}\mu(E_{k})
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f,g\in{\cal S}(\Omega)^{+}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\alpha\in[0,+\infty)$
+\end_inset
+
+, 
+\begin_inset Formula $\int\alpha f\,d\mu=\alpha\int f\,d\mu$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Supongamos 
+\begin_inset Formula $f=:\sum_{i=1}^{n}a_{i}\chi_{A_{i}}$
+\end_inset
+
+ y 
+\begin_inset Formula $g=:\sum_{j=1}^{m}b_{j}\chi_{B_{j}}$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $\alpha f=\sum_{i=1}^{n}\alpha a_{i}\chi_{A_{i}}$
+\end_inset
+
+ y
+\begin_inset Formula 
+\[
+\int\alpha f\,d\mu=\sum_{i=1}^{n}\alpha a_{i}\mu(A_{i})=\alpha\sum_{i=1}^{n}a_{i}\mu(A_{i})=\alpha\int f\,d\mu
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\int(f+g)d\mu=\int f\,d\mu+\int g\,d\mu$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Sean 
+\begin_inset Formula $C:=\{c_{1},\dots,c_{r}\}=\{a_{i}+b_{j}\}_{i\in\{1,\dots,n\},j\in\{1,\dots,m\}}$
+\end_inset
+
+, y 
+\begin_inset Formula $C_{k}:=\bigcup_{a_{i}+b_{j}=c_{k}}(A_{i}\cap B_{j})$
+\end_inset
+
+ para cada 
+\begin_inset Formula $c_{k}$
+\end_inset
+
+, tenemos 
+\begin_inset Formula $f+g=\sum_{k=1}^{r}c_{k}\chi_{C_{k}}$
+\end_inset
+
+.
+ Entonces
+\begin_inset Formula 
+\begin{multline*}
+\int f+g\,d\mu=\sum_{k=1}^{r}c_{k}\mu(C_{k})=\sum_{k=1}^{r}c_{k}\mu(C_{k})=\sum_{k=1}^{r}c_{k}\left(\sum_{a_{i}+b_{j}=c_{k}}\mu(A_{i}\cap B_{j})\right)=\\
+=\sum_{k=1}^{r}\sum_{a_{i}+b_{j}=c_{k}}(a_{i}+b_{j})\mu(A_{i}\cap B_{j})=\sum_{i=1}^{n}\sum_{j=1}^{m}((a_{i}+b_{j})\mu(A_{i}\cap B_{j})=\\
+=\sum_{i=1}^{n}a_{i}\left(\sum_{j=1}^{m}\mu(A_{i}\cap B_{j})\right)+\sum_{j=1}^{m}b_{j}\left(\sum_{i=1}^{n}\mu(A_{i}\cap B_{j})\right)=\sum_{i=1}^{n}a_{i}\mu(A_{i})+\sum_{j=1}^{m}b_{i}\mu(B_{j})
+\end{multline*}
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f\leq g$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\int f\,d\mu\leq\int g\,d\mu$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\int g\,d\mu=\int f\,d\mu+\int(g-f)d\mu\geq\int f\,d\mu
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\nu:\Sigma\rightarrow[0,+\infty]$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\nu(E):=\int f\chi_{E}d\mu$
+\end_inset
+
+ es una medida finita.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\nu(E)=\sum_{i=1}^{n}a_{i}\mu(A_{i}\cap E)$
+\end_inset
+
+, y como las aplicaciones 
+\begin_inset Formula $\nu_{i}(E)=\mu(E\cap A_{i})$
+\end_inset
+
+ son medidas, 
+\begin_inset Formula $\nu$
+\end_inset
+
+ también lo es.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Para 
+\begin_inset Formula $f:\Omega\rightarrow[0,+\infty]$
+\end_inset
+
+ medible, se define
+\begin_inset Formula 
+\[
+\int f\,d\mu:=\sup\left\{ \int s\,d\mu:s\in{\cal S}(\Omega)\land0\leq s\leq f\right\} 
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Esta definición es compatible con la dada inicialmente para funciones simples.
+ Para 
+\begin_inset Formula $E\in\Sigma$
+\end_inset
+
+, se define
+\begin_inset Formula 
+\[
+\int_{E}f\,d\mu:=\int\chi_{E}f\,d\mu
+\]
+
+\end_inset
+
+
+\begin_inset Newpage pagebreak
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $f,g:\Omega\rightarrow[0,+\infty]$
+\end_inset
+
+ medibles y 
+\begin_inset Formula $A,B\in\Sigma$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $f\leq g\implies\int f\,d\mu\leq\int g\,d\mu$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $A\subseteq B\implies\int_{A}f\,d\mu\leq\int_{B}f\,d\mu$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall\alpha\in[0,+\infty),\int\alpha f\,d\mu=\alpha\int f\,d\mu$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $f|_{A}=0\implies\int_{A}f\,d\mu=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\mu(A)=0\implies\int_{A}f\,d\mu=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+La 
+\series bold
+desigualdad de Tchevichev
+\series default
+ afirma que si 
+\begin_inset Formula $f:\Omega\rightarrow[0,+\infty]$
+\end_inset
+
+ es medible y 
+\begin_inset Formula $E\in\Sigma$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Enumerate
+Si para 
+\begin_inset Formula $t>0$
+\end_inset
+
+ definimos 
+\begin_inset Formula $E_{t}:=E\cap\{f>t\}$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+t\mu(E_{t})\leq\int_{E_{t}}f\,d\mu\leq\int f\,d\mu
+\]
+
+\end_inset
+
+Se obtiene de integrar en 
+\begin_inset Formula $t\chi_{E_{t}}\leq f\chi_{E_{t}}\leq f$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\int_{E}f\,d\mu=0$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f(\omega)=0$
+\end_inset
+
+ para casi todo 
+\begin_inset Formula $\omega\in E$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Por lo anterior, 
+\begin_inset Formula $\forall k\in\mathbb{N},\mu(E_{\frac{1}{k}})=0$
+\end_inset
+
+, y como 
+\begin_inset Formula $\{f\neq0\}=\bigcup_{k}E_{\frac{1}{k}}$
+\end_inset
+
+, este conjunto también tiene medida nula.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\int_{E}f\,d\mu<+\infty$
+\end_inset
+
+ entonces 
+\begin_inset Formula $f(\omega)<+\infty$
+\end_inset
+
+ para casi todo 
+\begin_inset Formula $\omega\in E$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\mu(E_{k})\leq\frac{1}{k}\int f\,d\mu$
+\end_inset
+
+, luego 
+\begin_inset Formula $\mu(\{f=+\infty\})=\mu\left(\bigcap_{k}E_{k}\right)=\lim_{n}\mu(E_{k})=0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $\int_{E}f\,d\mu<+\infty$
+\end_inset
+
+ entonces 
+\begin_inset Formula $\{f>0\}$
+\end_inset
+
+ es 
+\begin_inset Formula $\sigma$
+\end_inset
+
+-finito.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+\begin_inset Formula $\{f>0\}=\bigcup_{k}E_{\frac{1}{k}}$
+\end_inset
+
+ y cada 
+\begin_inset Formula $E_{\frac{1}{k}}$
+\end_inset
+
+ tiene medida finita.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+El 
+\series bold
+teorema de convergencia monótona de Lebesgue
+\series default
+ afirma que si 
+\begin_inset Formula $(f_{n}:\Omega\rightarrow[0,+\infty])_{n}$
+\end_inset
+
+ es una sucesión creciente de funciones medibles que converge puntualmente
+ a 
+\begin_inset Formula $f:\Omega\rightarrow[0,+\infty]$
+\end_inset
+
+, entonces
+\begin_inset Formula 
+\[
+\int f\,d\mu=\lim_{n}\int f_{n}d\mu
+\]
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $\alpha:=\sup\int f_{n}d\mu\in[0,+\infty]$
+\end_inset
+
+, como 
+\begin_inset Formula $f_{n}\leq f\forall n$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ es medible por ser límite puntual de medibles, 
+\begin_inset Formula $\alpha\leq\int f$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $s\leq f$
+\end_inset
+
+ una función simple, al ser 
+\begin_inset Formula $f_{n}$
+\end_inset
+
+ creciente y convergente a 
+\begin_inset Formula $f$
+\end_inset
+
+, se tiene que 
+\begin_inset Formula $(E_{n}:=\{f_{n}\geq s\})_{n}$
+\end_inset
+
+ es creciente con 
+\begin_inset Formula $\bigcup_{n}E_{n}=\Omega$
+\end_inset
+
+.
+ Tenemos que 
+\begin_inset Formula $\int f_{n}d\mu\geq\int\chi_{E_{n}}f_{n}d\mu\geq\int\chi_{E_{n}}s\,d\mu$
+\end_inset
+
+.
+ Tomando límites aquí, vemos que 
+\begin_inset Formula $\alpha\geq\int\chi_{\Omega}s\,d\mu=\int s\,d\mu$
+\end_inset
+
+, y tomando supremos entre las funciones simples 
+\begin_inset Formula $s\leq f$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha\geq\int f\,d\mu$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Usando esto podemos probar para 
+\begin_inset Formula $f,g:\Omega\rightarrow[0,+\infty]$
+\end_inset
+
+ que 
+\begin_inset Formula $\int(f+g)d\mu=\int f\,d\mu+\int g\,d\mu$
+\end_inset
+
+, partiendo de la propiedad correspondiente para funciones simples.
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+teorema de Beppo-Levi
+\series default
+ afirma que la suma de una sucesión de funciones medibles 
+\begin_inset Formula $f_{n}:\Omega\rightarrow[0,+\infty]$
+\end_inset
+
+ es medible y
+\begin_inset Formula 
+\[
+\int\left(\sum_{n=1}^{\infty}f_{n}\right)d\mu=\sum_{n=1}^{\infty}\int f_{n}d\mu
+\]
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $(F_{n}:=\sum_{k=1}^{n}f_{k})_{n}$
+\end_inset
+
+ es una sucesión de funciones medibles, por lo que basta tomar límites en
+ 
+\begin_inset Formula $\int F_{n}d\mu=\sum_{k=1}^{n}\int f_{k}d\mu$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Una función 
+\begin_inset Formula $\Omega\rightarrow[0,+\infty]$
+\end_inset
+
+ medible es 
+\series bold
+integrable
+\series default
+ si su integral es finita.
+ Por ejemplo, si 
+\begin_inset Formula $\Sigma={\cal P}(\Omega)$
+\end_inset
+
+ y 
+\begin_inset Formula $\mu(E)=|E|\forall E\in\Sigma$
+\end_inset
+
+, una función 
+\begin_inset Formula $f$
+\end_inset
+
+ medible positiva es integrable si y sólo si 
+\begin_inset Formula $\int f\,d\mu=\sum_{\omega\in\Omega}f(\omega)<+\infty$
+\end_inset
+
+, y si además 
+\begin_inset Formula $\Omega=\mathbb{N}$
+\end_inset
+
+, la sucesión 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable si y sólo si la serie 
+\begin_inset Formula $\sum_{n}f(n)$
+\end_inset
+
+ converge, y la integral coincide con la suma.
+ En tal caso, el teorema de Beppo-Levi nos dice que si 
+\begin_inset Formula $a_{n,m}$
+\end_inset
+
+ es una sucesión doble de números positivos, entonces
+\begin_inset Formula 
+\[
+\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}a_{m,n}=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}a_{m,n}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+lema de Fatou
+\series default
+ afirma que si 
+\begin_inset Formula $f_{n}:\Omega\rightarrow[0,+\infty]$
+\end_inset
+
+ es una sucesión de funciones medibles, su límite inferior 
+\begin_inset Formula $f(\omega):=\liminf_{n}f_{n}(\omega)$
+\end_inset
+
+ es medible y 
+\begin_inset Formula $\int f\,d\mu\leq\liminf_{n}\int f_{n}d\mu$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $(g_{n}:=\inf_{m\geq n}\{f_{m}(x)\})_{n}$
+\end_inset
+
+ define una sucesión creciente de funciones medibles convergente hacia 
+\begin_inset Formula $f$
+\end_inset
+
+, y por la monotonía de la integral y el teorema de convergencia monótona,
+ 
+\begin_inset Formula 
+\[
+\int f\,d\mu=\sup_{n\in\mathbb{N}}\left\{ \int g_{n}d\mu\right\} \leq\sup_{n\in\mathbb{N}}\left\{ \inf_{m\geq n}\left\{ \int f_{m}d\mu\right\} \right\} =\liminf_{n}\int f_{n}d\mu
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $g:\Omega\rightarrow[0,+\infty]$
+\end_inset
+
+ es medible, entonces 
+\begin_inset Formula $\nu:\Sigma\rightarrow[0,+\infty]$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\nu(E):=\int_{E}g\,d\mu$
+\end_inset
+
+ es una medida y para 
+\begin_inset Formula $f:\Omega\rightarrow[0,+\infty]$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\int f\,d\nu=\int fg\,d\mu
+\]
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ 
+\begin_inset Formula $\nu(\emptyset)=0$
+\end_inset
+
+ y, dada una sucesión 
+\begin_inset Formula $(A_{n})_{n}$
+\end_inset
+
+ de medibles disjuntos, 
+\begin_inset Formula $g\chi_{\bigcup_{n}A_{n}}=\sum_{n}g\chi_{A_{n}}$
+\end_inset
+
+ y el teorema de Beppo-Levi nos da que 
+\begin_inset Formula $\nu(\bigcup_{n}A_{n})=\sum_{n}\nu(A_{n})$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $\nu$
+\end_inset
+
+ es una medida.
+ Sea 
+\begin_inset Formula $s:=\sum_{i=1}^{m}a_{i}\chi_{E_{i}}\in{\cal S}(\Omega)^{+}$
+\end_inset
+
+,
+\begin_inset Formula 
+\[
+\int s\,d\nu=\sum_{i=1}^{m}a_{i}\int\chi_{E_{i}}g\,d\mu=\int\left(\sum_{i=1}^{m}a_{i}\chi_{E_{i}}\right)g\,d\mu=\int sg\,d\mu
+\]
+
+\end_inset
+
+Usando el teorema de la convergencia monótona para tomar límites en una
+ sucesión creciente de funciones simples que converge a 
+\begin_inset Formula $f$
+\end_inset
+
+ se completa la prueba.
+\end_layout
+
+\begin_layout Section
+Funciones integrables
+\end_layout
+
+\begin_layout Standard
+Una función medible 
+\begin_inset Formula $f:\Omega\rightarrow\mathbb{R}$
+\end_inset
+
+ es integrable si 
+\begin_inset Formula $\int|f|d\mu<+\infty$
+\end_inset
+
+, si y sólo si 
+\begin_inset Formula $f^{+}=\max\{f,0\},f^{-}=\min\{f,0\}:\Omega\rightarrow[-\infty,+\infty]$
+\end_inset
+
+ son integrables, y definimos
+\begin_inset Formula 
+\[
+\int f\,d\mu:=\int f^{+}\,d\mu-\int f^{-}\,d\mu
+\]
+
+\end_inset
+
+Si solo una de entre 
+\begin_inset Formula $f^{+}$
+\end_inset
+
+ y 
+\begin_inset Formula $f^{-}$
+\end_inset
+
+ es integrable, usamos esta misma definición y entonces 
+\begin_inset Formula $\int f\,d\mu\in[-\infty,+\infty]$
+\end_inset
+
+.
+ Una función medible 
+\begin_inset Formula $f:\Omega\rightarrow\mathbb{C}$
+\end_inset
+
+ es integrable si 
+\begin_inset Formula $\int|f|d\mu<+\infty$
+\end_inset
+
+.
+ Si 
+\begin_inset Formula $u,v:\Omega\rightarrow\mathbb{R}$
+\end_inset
+
+ son tales que 
+\begin_inset Formula $f=u+iv$
+\end_inset
+
+, es claro que 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable si y sólo si 
+\begin_inset Formula $u$
+\end_inset
+
+ y 
+\begin_inset Formula $v$
+\end_inset
+
+ lo son, y definimos
+\begin_inset Formula 
+\[
+\int f\,d\mu:=\int u\,d\mu+i\int v\,d\mu
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula ${\cal L}^{1}(\Omega,\Sigma,\mu)$
+\end_inset
+
+ al conjunto de funciones integrables 
+\begin_inset Formula $f:\Omega\rightarrow\mathbb{C}$
+\end_inset
+
+.
+ Cuando basta indicar uno de los componentes de la terna y se sobreentiende
+ el resto, podemos escribir 
+\begin_inset Formula ${\cal L}^{1}(\Omega)$
+\end_inset
+
+, 
+\begin_inset Formula ${\cal L}^{1}(\Sigma)$
+\end_inset
+
+ o 
+\begin_inset Formula ${\cal L}^{1}(\mu)$
+\end_inset
+
+.
+ El subconjunto de 
+\begin_inset Formula ${\cal L}^{1}$
+\end_inset
+
+ formado por las funciones reales se denota 
+\begin_inset Formula ${\cal L}_{\mathbb{R}}^{1}(\mu)$
+\end_inset
+
+, y por definición 
+\begin_inset Formula $f\in{\cal L}^{1}(\mu)\iff|f|\in{\cal L}^{1}(\mu)$
+\end_inset
+
+.
+ Propiedades:
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula ${\cal L}^{1}(\mu)$
+\end_inset
+
+ es un espacio vectorial complejo y, si lo consideramos como un espacio
+ vectorial real por el isomorfismo canónico entre 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ y 
+\begin_inset Formula $\mathbb{R}^{2}$
+\end_inset
+
+, 
+\begin_inset Formula ${\cal L}_{\mathbb{R}}^{1}$
+\end_inset
+
+ es un subespacio suyo.
+\begin_inset Newline newline
+\end_inset
+
+Sean 
+\begin_inset Formula $f,g\in{\cal L}^{1}(\mu)$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha\in\mathbb{C}$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha f+g$
+\end_inset
+
+ es medible y 
+\begin_inset Formula $|\alpha f+g|\leq|\alpha||f|+|g|$
+\end_inset
+
+, e integrando y aplicando las propiedades de la integral en funciones positivas
+, 
+\begin_inset Formula $\int|\alpha f+g|d\mu\leq\int(|\alpha||f|+|g|)d\mu=|\alpha|\int|f|d\mu+\int|g|d\mu<+\infty$
+\end_inset
+
+,
+\end_layout
+
+\begin_layout Enumerate
+La aplicación 
+\begin_inset Formula $\nu:{\cal L}^{1}(\mu)\rightarrow\mathbb{C}$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\nu(f):=\int f\,d\mu$
+\end_inset
+
+ es lineal.
+\begin_inset Newline newline
+\end_inset
+
+Sean 
+\begin_inset Formula $u,v\in{\cal L}_{\mathbb{R}}^{1}(\mu)$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha\in\mathbb{R}$
+\end_inset
+
+:
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Formula $i\int(u+iv)d\mu=i\int ud\mu+i^{2}\int v\,d\mu=\int(-v+iu)d\mu=\int i(u+iv)d\mu$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $h:=u+v\implies h^{+}-h^{-}=u^{+}-u^{-}+v^{+}-v^{-}\implies h^{+}+(u^{-}+v^{-})=(u^{+}+v^{+})+h^{-}\implies\int h^{+}d\mu+\int u^{-}d\mu+\int v^{-}d\mu=\int u^{+}d\mu+\int v^{+}d\mu+\int h^{-}d\mu\implies\int h\,d\mu=\int h^{+}d\mu-\int h^{-}d\mu=\int u^{+}d\mu-\int u^{-}d\mu+\int v^{+}d\mu-\int v^{-}d\mu=\int u\,d\mu+\int v\,d\mu$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\alpha\geq0\implies(\alpha u)^{+}=\alpha u^{+}\land(\alpha u)^{-}=\alpha u^{-}\implies\int\alpha u\,d\mu=\int\alpha u^{+}d\mu-\int\alpha u^{-}d\mu=\alpha\left(\int u^{+}d\mu-\int u^{-}d\mu\right)=\alpha\int u\,d\mu$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\alpha<0\implies(\alpha u)^{+}=(-\alpha)u^{-}\land(\alpha u)^{-}=(-\alpha)u^{+}\implies\int\alpha u\,d\mu=\int(-\alpha)u^{-}d\mu-\int(-\alpha)u^{+}d\mu=(-\alpha)\left(\int u^{-}d\mu-\int u^{+}d\mu\right)=\alpha\int u\,d\mu$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Desigualdad triangular:
+\begin_inset Formula 
+\[
+\forall f\in{\cal L}^{1}(\mu),\left|\int f\,d\mu\right|\leq\int|f|d\mu
+\]
+
+\end_inset
+
+Si 
+\begin_inset Formula $\left|\int f\,d\mu\right|=0$
+\end_inset
+
+ es trivial.
+ Si 
+\begin_inset Formula $\left|\int f\,d\mu\right|\neq0$
+\end_inset
+
+, sea 
+\begin_inset Formula $a:=\frac{\int f\,d\mu}{|\int f\,d\mu|}$
+\end_inset
+
+ y como 
+\begin_inset Formula $\forall z\in\mathbb{C},|\text{Re}(z)|\leq|z|$
+\end_inset
+
+, 
+\begin_inset Formula $|\text{Re}(af)|\leq|af|=|f|$
+\end_inset
+
+; de aquí que 
+\begin_inset Formula $\left|\int f\,d\mu\right|=a\int f\,d\mu=\text{Re}\left(\int af\,d\mu\right)=\int\text{Re}(af)d\mu\leq\int|f|d\mu$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Una función medible 
+\begin_inset Formula $f:E\rightarrow\mathbb{C}$
+\end_inset
+
+ definida en 
+\begin_inset Formula $E\in\Sigma$
+\end_inset
+
+ es 
+\series bold
+integrable sobre 
+\begin_inset Formula $E$
+\end_inset
+
+
+\series default
+ si la 
+\series bold
+extensión canónica
+\series default
+ 
+\begin_inset Formula $\tilde{f}:=f\chi_{E}$
+\end_inset
+
+ es integrable, y entonces se define 
+\begin_inset Formula $\int_{E}f\,d\mu:=\int\tilde{f}\,d\mu$
+\end_inset
+
+.
+ Con esto:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f:E\rightarrow\mathbb{C}$
+\end_inset
+
+ es una función medible definida en 
+\begin_inset Formula $E\in\Sigma$
+\end_inset
+
+, 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable sobre 
+\begin_inset Formula $E$
+\end_inset
+
+ y 
+\begin_inset Formula $\forall A\in\Sigma\cap{\cal P}(E),\int_{A}f\,d\mu=0$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $f$
+\end_inset
+
+ se anula en casi todo 
+\begin_inset Formula $\omega\in E$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+En particular, 
+\begin_inset Formula $\int(\text{Re}(f))^{+}d\mu=\text{Re}\left(\int_{\{\text{Re}(f)\geq0\}}f\,d\mu\right)=0$
+\end_inset
+
+, y por la desigualdad de Tchevichev, 
+\begin_inset Formula $\mu\{\text{Re}(f)^{+}>t\}\leq\frac{1}{t}\int(\text{Re}(f))^{+}d\mu=0$
+\end_inset
+
+ para todo 
+\begin_inset Formula $t>0$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\text{Re}(f)^{+}$
+\end_inset
+
+ se anula en casi todo punto.
+ Lo mismo sucede con 
+\begin_inset Formula $(\text{Re}(f))^{-}$
+\end_inset
+
+, 
+\begin_inset Formula $(\text{Im}(f))^{+}$
+\end_inset
+
+ e 
+\begin_inset Formula $(\text{Im}(f))^{-}$
+\end_inset
+
+.
+ Por tanto 
+\begin_inset Formula $f$
+\end_inset
+
+ se anula en casi todo punto.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+Para 
+\begin_inset Formula $A\in\Sigma\cap{\cal P}(E)$
+\end_inset
+
+, 
+\begin_inset Formula $(\text{Re}(f))^{+}$
+\end_inset
+
+, 
+\begin_inset Formula $(\text{Re}(f))^{-}$
+\end_inset
+
+, 
+\begin_inset Formula $(\text{Im}(f))^{+}$
+\end_inset
+
+ e 
+\begin_inset Formula $(\text{Im}(f))^{-}$
+\end_inset
+
+ se anulan en casi todo 
+\begin_inset Formula $\omega\in A\subseteq E$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\text{Re}\left(\int f\,d\mu\right)=\int\text{Re}(f)d\mu=\int(\text{Re}(f))^{+}d\mu-\int(\text{Re}(f))^{-}d\mu=0-0=0$
+\end_inset
+
+ y, análogamente, 
+\begin_inset Formula $\text{Im}\left(\int f\,d\mu\right)=0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f,g:E\rightarrow\mathbb{C}$
+\end_inset
+
+ son medibles iguales en casi todo punto y 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable sobre 
+\begin_inset Formula $E$
+\end_inset
+
+ entonces 
+\begin_inset Formula $g$
+\end_inset
+
+ también lo es y 
+\begin_inset Formula $\int_{E}f\,d\mu=\int_{E}g\,d\mu$
+\end_inset
+
+.
+\begin_inset Newline newline
+\end_inset
+
+Basta aplicar el punto anterior a 
+\begin_inset Formula $f-g$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+El 
+\series bold
+teorema de la convergencia dominada
+\series default
+ afirma que, dada una sucesión 
+\begin_inset Formula $(f_{n}:\Omega\rightarrow\mathbb{C})_{n}$
+\end_inset
+
+ de funciones que converge en casi todo punto, si existe 
+\begin_inset Formula $g:\Omega\rightarrow\mathbb{C}$
+\end_inset
+
+ integrable con 
+\begin_inset Formula $|f_{n}|\leq g\forall n\in\mathbb{N}$
+\end_inset
+
+, entonces la función límite 
+\begin_inset Formula $f(\omega):=\lim_{n}f_{n}(\omega)$
+\end_inset
+
+, definida en casi todo punto, es integrable, 
+\begin_inset Formula 
+\[
+\lim_{n}\int|f_{n}-f|d\mu=0
+\]
+
+\end_inset
+
+y, en particular, 
+\begin_inset Formula 
+\[
+\lim_{n}\int f_{n}d\mu=\int f\,d\mu
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Las 
+\begin_inset Formula $f_{n}$
+\end_inset
+
+ son integrables al estar acotadas por una función integrable.
+ Te
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+ne
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+mos que 
+\begin_inset Formula $|f-f_{n}|\leq|f|+|f_{n}|\leq2g$
+\end_inset
+
+ y, teniendo en cuenta que 
+\begin_inset Formula $\limsup_{n}|f-f_{n}|=0$
+\end_inset
+
+ y aplicando el lema de Fatou a la sucesión de medibles positivas 
+\begin_inset Formula $2g-|f-f_{n}|$
+\end_inset
+
+, queda que 
+\begin_inset Formula $\int2g\,d\mu=\int\liminf_{n}(2g-|f-f_{n}|)d\mu\leq\liminf_{n}\int(2g-|f-f_{n}|)d\mu=\int2g\,d\mu-\limsup_{n}\int|f-f_{n}|$
+\end_inset
+
+.
+ Restando la cantidad finita 
+\begin_inset Formula $\int2g\,d\mu$
+\end_inset
+
+ a ambos miembros de la desigualdad, 
+\begin_inset Formula $0\leq-\limsup_{n}\int|f-f_{n}|\leq0$
+\end_inset
+
+, con lo que 
+\begin_inset Formula $\lim_{n}\int|f-f_{n}|=0$
+\end_inset
+
+ y 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable porque 
+\begin_inset Formula $|f|\leq|f_{n}|+|f-f_{n}|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{sloppypar}
+\end_layout
+
+\end_inset
+
+Como 
+\series bold
+teorema
+\series default
+, dada una sucesión de funciones integrables 
+\begin_inset Formula $(f_{n}:\Omega\rightarrow\mathbb{C})_{n}$
+\end_inset
+
+ con 
+\begin_inset Formula $\sum_{n=1}^{\infty}\int|f_{n}|d\mu<+\infty$
+\end_inset
+
+, la serie 
+\begin_inset Formula $\sum_{n=1}^{\infty}f_{n}(\omega)$
+\end_inset
+
+ converge absolutamente en casi todo punto y su suma es integrable con 
+\begin_inset Formula $\int\left(\sum_{n=1}^{\infty}f_{n}\right)d\mu=\sum_{n=1}^{\infty}\int f_{n}d\mu$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Por el teorema de la convergencia monótona, 
+\begin_inset Formula $G:=\sum_{n}|f_{n}|$
+\end_inset
+
+ converge en casi todo punto y es integrable, y 
+\begin_inset Formula $S:=\sum_{n}f_{n}$
+\end_inset
+
+ también, y como para 
+\begin_inset Formula $(g_{m}:=\sum_{n=0}^{m}f_{n})_{m}$
+\end_inset
+
+ se tiene 
+\begin_inset Formula $|g_{m}|\leq G$
+\end_inset
+
+, podemos aplicar el teorema de la convergencia dominada para obtener el
+ resultado.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{sloppypar}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Relación entre las integrales de Riemann y Lebesgue
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, las funciones integrables Riemann son integrables respecto a la medida
+ de Lebesgue y las integrales coinciden.
+ 
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $f:[a,b]\rightarrow\mathbb{R}$
+\end_inset
+
+ integrable Riemann, y por tanto acotada, existe una sucesión 
+\begin_inset Formula $(P_{k})_{k}$
+\end_inset
+
+ de particiones en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ con 
+\begin_inset Formula $\Vert P_{n}\Vert\rightarrow0$
+\end_inset
+
+ y 
+\begin_inset Formula $\int_{[a,b]}f(\vec{x})d\vec{x}=\lim_{k}s(f,P_{k})$
+\end_inset
+
+, y si 
+\begin_inset Formula $\{N_{k1},\dots,N_{km_{k}}\}$
+\end_inset
+
+ es el conjunto de subrectángulos de 
+\begin_inset Formula $P_{k}$
+\end_inset
+
+, la sucesión 
+\begin_inset Formula $(s_{k})_{k}$
+\end_inset
+
+ de funciones simples 
+\begin_inset Formula $s_{k}(t):=\sum_{i=1}^{m_{k}}\chi_{N_{ki}}\inf_{x\in N_{ki}}\{f(x)\}$
+\end_inset
+
+ está acotada por la función constante 
+\begin_inset Formula $M:=\chi_{[a,b]}\sup_{x\in[a,b]}\{|f(x)|\}$
+\end_inset
+
+ y converge a 
+\begin_inset Formula $f$
+\end_inset
+
+ en todos sus puntos de continuidad y por tanto en casi todo punto.
+ Entonces, por el teorema de la convergencia dominada, 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable Lebesgue en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ y 
+\begin_inset Formula $\int_{[a,b]}f\,d\lambda_{n}=\int_{[a,b]}f(\vec{x})d\vec{x}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Una función 
+\begin_inset Formula $f:S\rightarrow\mathbb{R}$
+\end_inset
+
+ definida en un intervalo 
+\begin_inset Formula $n$
+\end_inset
+
+-dimensional 
+\begin_inset Formula $S\subseteq\mathbb{R}^{n}$
+\end_inset
+
+ se dice 
+\series bold
+localmente integrable Riemann
+\series default
+ si es integrable Riemann sobre cada intervalo compacto 
+\begin_inset Formula $I\subseteq S$
+\end_inset
+
+.
+ Definimos entonces la 
+\series bold
+integral impropia
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ en 
+\begin_inset Formula $S$
+\end_inset
+
+ como
+\begin_inset Formula 
+\[
+\int_{S}f(x)dx:=\lim_{I\nearrow S}\int_{I}f(x)dx
+\]
+
+\end_inset
+
+y si existe, decimos que 
+\series bold
+converge
+\series default
+.
+ La integral impropia 
+\begin_inset Formula $\int_{S}f(x)dx$
+\end_inset
+
+ es 
+\series bold
+absolutamente convergente
+\series default
+ si 
+\begin_inset Formula $\int_{S}|f(x)|dx$
+\end_inset
+
+ es convergente.
+\end_layout
+
+\begin_layout Standard
+Para 
+\begin_inset Formula $f:S\rightarrow\mathbb{R}$
+\end_inset
+
+ localmente integrable Riemann sobre un intervalo 
+\begin_inset Formula $n$
+\end_inset
+
+-dimensional 
+\begin_inset Formula $S\subseteq\mathbb{R}^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $\int_{S}f(x)dx$
+\end_inset
+
+ es absolutamente convergente si y sólo si 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable Lebesgue sobre 
+\begin_inset Formula $S$
+\end_inset
+
+ y 
+\begin_inset Formula $\int_{S}f\,d\lambda_{n}=\int_{S}f(x)dx$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\implies]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable Riemann, y por tanto Lebesgue, sobre todo intervalo compacto
+ 
+\begin_inset Formula $I\subseteq S$
+\end_inset
+
+, y dada una sucesión 
+\begin_inset Formula $(I_{k})_{k}$
+\end_inset
+
+ de intervalos compactos en 
+\begin_inset Formula $S$
+\end_inset
+
+ que tiende a 
+\begin_inset Formula $S$
+\end_inset
+
+, 
+\begin_inset Formula $(f_{k}:=\chi_{I_{k}}f)$
+\end_inset
+
+ es una sucesión de funciones que tiende a 
+\begin_inset Formula $f$
+\end_inset
+
+ tal que 
+\begin_inset Formula $|f_{k}|<|f|\forall k\in\mathbb{N}$
+\end_inset
+
+, y por la convergencia dominada tenemos que 
+\begin_inset Formula $\int_{S}f\,d\lambda_{n}=\lim_{k}\int f_{k}d\lambda_{n}=\lim_{k}\int_{I_{k}}f\,dx=\int_{S}f(x)dx$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Argument item:1
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\impliedby]$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{sloppypar}
+\end_layout
+
+\end_inset
+
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable Riemann en un intervalo compacto 
+\begin_inset Formula $I\subseteq S$
+\end_inset
+
+, también es acotada en ese intervalo y por tanto 
+\begin_inset Formula $f^{+}$
+\end_inset
+
+ y 
+\begin_inset Formula $f^{-}$
+\end_inset
+
+ también lo son.
+ Si además 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable Lebesgue sobre 
+\begin_inset Formula $S$
+\end_inset
+
+, por la convergencia monótona tenemos que si 
+\begin_inset Formula $(I_{k})_{k}$
+\end_inset
+
+ es una sucesión creciente de in
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+ter
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+va
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+-
+\end_layout
+
+\end_inset
+
+los compactos que tiende a 
+\begin_inset Formula $S$
+\end_inset
+
+, 
+\begin_inset Formula $\int_{S}f^{+}(x)dx=\lim_{I\nearrow S}\int_{I}f^{+}(x)dx=\lim_{k}\int_{I_{k}}f^{+}(x)dx=\lim_{k}\int_{I_{k}}f^{+}(x)d\lambda_{n}=\int\lim_{k}\chi_{I_{k}}f^{+}(x)d\lambda_{n}=\int_{S}f^{+}(x)d\lambda_{n}<+\infty$
+\end_inset
+
+, y análogamente 
+\begin_inset Formula $\int_{S}f^{-}(x)dx<+\infty$
+\end_inset
+
+, por lo que 
+\begin_inset Formula $\int_{S}|f(x)|dx=\int_{S}f^{+}(x)dx+\int_{S}f^{-}(x)dx<+\infty$
+\end_inset
+
+.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{sloppypar}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+soporte
+\series default
+ de una función 
+\begin_inset Formula $g:\Omega\rightarrow\mathbb{C}$
+\end_inset
+
+ a 
+\begin_inset Formula $\text{sop}(g):=\overline{\{g\neq0\}}$
+\end_inset
+
+, y 
+\begin_inset Formula ${\cal C}_{0}(\Omega)$
+\end_inset
+
+ al conjunto de funciones continuas 
+\begin_inset Formula $g:\Omega\rightarrow\mathbb{C}$
+\end_inset
+
+ con soporte compacto.
+ Como 
+\series bold
+teorema
+\series default
+, 
+\begin_inset Formula ${\cal C}_{0}(\mathbb{R}^{n})$
+\end_inset
+
+ es denso en 
+\begin_inset Formula ${\cal L}^{1}(\mathbb{R}^{n})$
+\end_inset
+
+.
+ 
+\series bold
+Demostración:
+\series default
+ Como las funciones simples medibles son densas en 
+\begin_inset Formula ${\cal L}^{1}(\mathbb{R}^{n})$
+\end_inset
+
+, basta ver que la función característica de cualquier conjunto medible
+ Lebesgue es límite de una sucesión de funciones en 
+\begin_inset Formula ${\cal C}_{0}(\mathbb{R}^{n})$
+\end_inset
+
+.
+ Para ello, dado un medible 
+\begin_inset Formula $E$
+\end_inset
+
+ y 
+\begin_inset Formula $\varepsilon>0$
+\end_inset
+
+, existen 
+\begin_inset Formula $K$
+\end_inset
+
+ compacto y 
+\begin_inset Formula $A$
+\end_inset
+
+ abierto con 
+\begin_inset Formula $K\subseteq E\subseteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $\lambda(A\backslash K)<\frac{\varepsilon}{2}$
+\end_inset
+
+, y si 
+\begin_inset Formula $f\in{\cal C}_{0}(\mathbb{R}^{n})$
+\end_inset
+
+ cumple que 
+\begin_inset Formula $K\subseteq\text{sop}(f)\subseteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $\chi_{K}\leq f\leq1$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\Vert f-\chi_{E}\Vert_{\infty}\leq2\chi_{A\backslash K}$
+\end_inset
+
+ y 
+\begin_inset Formula $\Vert f-\chi_{E}\Vert_{\infty}\leq2\lambda(A\backslash K)<\varepsilon$
+\end_inset
+
+.
+ Para ver que existe 
+\begin_inset Formula $f$
+\end_inset
+
+, fijado un cerrado 
+\begin_inset Formula $F\subseteq\mathbb{R}^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $d(x,F)$
+\end_inset
+
+ es continua, y como 
+\begin_inset Formula $\delta:=\min\{d(x,K):x\notin A\}>0$
+\end_inset
+
+, 
+\begin_inset Formula $A_{0}:=\{x:d(x,K)<\frac{\delta}{2}\}$
+\end_inset
+
+ es un abierto acotado con 
+\begin_inset Formula $K\subseteq A_{0}\subseteq\overline{A_{0}}\subseteq A$
+\end_inset
+
+.
+ Tomando 
+\begin_inset Formula $F_{0}:=\mathbb{R}^{n}\backslash A_{0}=\{x:d(x,K)\geq\frac{\delta}{2}\}$
+\end_inset
+
+, podemos definir 
+\begin_inset Formula $f_{0}(y):=\frac{d(y,F_{0})}{1+d(y,F_{0})}$
+\end_inset
+
+, que cumple 
+\begin_inset Formula $0\leq f_{0}\leq1$
+\end_inset
+
+, 
+\begin_inset Formula $f_{0}(x)=0\forall x\in F_{0}\supseteq\mathbb{R}^{n}\backslash A$
+\end_inset
+
+ y, como para 
+\begin_inset Formula $y\in K$
+\end_inset
+
+, 
+\begin_inset Formula $d(y,F_{0})\geq\frac{\delta}{2}$
+\end_inset
+
+, 
+\begin_inset Formula $f_{0}(y)\geq k_{0}:=\frac{\frac{\delta}{2}}{1+\frac{\delta}{2}}$
+\end_inset
+
+ y la función continua 
+\begin_inset Formula $f(x):=\min\{1,\frac{f_{0}(x)}{k_{0}}\}$
+\end_inset
+
+ tiene soporte compacto en 
+\begin_inset Formula $\overline{A_{0}}\subseteq A$
+\end_inset
+
+ y 
+\begin_inset Formula $\chi_{K}\leq f\leq1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Integrales dependientes de un parámetro
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+ sean 
+\begin_inset Formula $(\Omega,\Sigma,\mu)$
+\end_inset
+
+ un espacio de medida, 
+\begin_inset Formula $(X,d)$
+\end_inset
+
+ un espacio métrico y 
+\begin_inset Formula $f:X\times\Omega\rightarrow\mathbb{C}$
+\end_inset
+
+ una función tal que:
+\end_layout
+
+\begin_layout Enumerate
+Para casi todo 
+\begin_inset Formula $\omega\in\Omega$
+\end_inset
+
+ 
+\begin_inset Formula $f^{\omega}(x):=f(x,\omega)$
+\end_inset
+
+ es continua.
+\end_layout
+
+\begin_layout Enumerate
+Para todo 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ 
+\begin_inset Formula $f_{x}(\omega):=f(x,\omega)$
+\end_inset
+
+ es medible.
+\end_layout
+
+\begin_layout Enumerate
+Existe 
+\begin_inset Formula $g:\Omega\rightarrow[0,+\infty)$
+\end_inset
+
+ tal que para todo 
+\begin_inset Formula $x\in X$
+\end_inset
+
+, para casi todo 
+\begin_inset Formula $\omega\in\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $|f(x,\omega)|\leq g(\omega)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Entonces para todo 
+\begin_inset Formula $x\in X$
+\end_inset
+
+ 
+\begin_inset Formula $f_{x}$
+\end_inset
+
+ es integrable y 
+\begin_inset Formula $F(x):=\int f(x,\omega)d\mu(\omega)$
+\end_inset
+
+ es continua.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Como 
+\begin_inset Formula $(X,d)$
+\end_inset
+
+ es un espacio métrico, la continuidad de las funciones en 
+\begin_inset Formula $X$
+\end_inset
+
+ se puede caracterizar mediante sucesiones, por lo que basta tomar una sucesión
+ 
+\begin_inset Formula $(x_{n})_{n}$
+\end_inset
+
+ de elementos de 
+\begin_inset Formula $X$
+\end_inset
+
+ que converge a un cierto 
+\begin_inset Formula $x$
+\end_inset
+
+ arbitrario y probar que 
+\begin_inset Formula 
+\[
+\left(F(x_{n})=\int f(x_{n},\omega)d\mu(\omega)\right)_{n}
+\]
+
+\end_inset
+
+ converge a 
+\begin_inset Formula $F(x)$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $N\in\Sigma$
+\end_inset
+
+ de medida cero tal que 
+\begin_inset Formula $f^{\omega}$
+\end_inset
+
+ es continua para 
+\begin_inset Formula $\omega\notin N$
+\end_inset
+
+ y por tanto 
+\begin_inset Formula $f^{\omega}(x_{n})=f(x_{n},\omega)\rightarrow f(x,\omega)$
+\end_inset
+
+.
+ Podemos tomar una sucesión 
+\begin_inset Formula $(A_{n})_{n}$
+\end_inset
+
+ de conjuntos de medida nula tales que para 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+ y 
+\begin_inset Formula $\omega\notin A_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $|f(x_{n},\omega)|\leq g(\omega)$
+\end_inset
+
+.
+ Entonces 
+\begin_inset Formula $N\cup\bigcup_{n}A_{n}$
+\end_inset
+
+ tiene medida nula y fuera de él, como la sucesión 
+\begin_inset Formula $(f_{x_{n}})_{n}$
+\end_inset
+
+ converge puntualmente y 
+\begin_inset Formula $|f_{x_{n}}|\leq g\forall n\in\mathbb{N}$
+\end_inset
+
+, podemos aplicar el teorema de la convergencia dominada y 
+\begin_inset Formula 
+\[
+\lim_{n}F(x_{n})=\lim_{n}\int f(x_{n},\omega)d\mu(\omega)=\int\lim_{n}f(x_{n},\omega)d\mu(\omega)=\int f(x,\omega)d\mu(\omega)=F(x)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Otro 
+\series bold
+teorema
+\series default
+ nos dice que si 
+\begin_inset Formula $I\subseteq\mathbb{R}$
+\end_inset
+
+ es un intervalo y 
+\begin_inset Formula $f:I\times\Omega\rightarrow\mathbb{C}$
+\end_inset
+
+ cumple que:
+\end_layout
+
+\begin_layout Enumerate
+Para casi todo 
+\begin_inset Formula $\omega\in\Omega$
+\end_inset
+
+ 
+\begin_inset Formula $f^{\omega}(x):=f(x,\omega)$
+\end_inset
+
+ es derivable (
+\begin_inset Formula $\exists\frac{\partial f}{\partial x}(x,\omega)$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Enumerate
+Para todo 
+\begin_inset Formula $x\in I$
+\end_inset
+
+ 
+\begin_inset Formula $f_{x}(\omega):=f(x,\omega)$
+\end_inset
+
+ es medible, siendo integrable para algún 
+\begin_inset Formula $x_{0}\in I$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Existe 
+\begin_inset Formula $g:\Omega\rightarrow[0,+\infty)$
+\end_inset
+
+ integrable tal que para casi todo 
+\begin_inset Formula $\omega\in\Omega$
+\end_inset
+
+ para todo 
+\begin_inset Formula $x\in I$
+\end_inset
+
+ 
+\begin_inset Formula $\left|\frac{\partial f}{\partial x}(x,\omega)\right|\leq g(\omega)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Entonces para todo 
+\begin_inset Formula $x\in I$
+\end_inset
+
+, 
+\begin_inset Formula $f_{x}$
+\end_inset
+
+ es integrable, 
+\begin_inset Formula $F(x):=\int f(x,\omega)d\mu(\omega)$
+\end_inset
+
+ es derivable y 
+\begin_inset Formula $F'(x)=\int\frac{\partial f}{\partial x}(x,\omega)d\mu(\omega)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+Demostración:
+\series default
+ Sea 
+\begin_inset Formula $A\in\Sigma$
+\end_inset
+
+ de medida nula tal que 
+\begin_inset Formula 
+\[
+\left|\frac{\partial f}{\partial x}(s,\omega)\right|\leq g(\omega)
+\]
+
+\end_inset
+
+ para 
+\begin_inset Formula $\omega\notin A$
+\end_inset
+
+ para todo 
+\begin_inset Formula $s\in I$
+\end_inset
+
+, por el teorema del incremento finito, si 
+\begin_inset Formula $\omega\notin A$
+\end_inset
+
+, 
+\begin_inset Formula 
+\[
+|f(x,\omega)-f(x_{0},\omega)|=\left|\frac{\partial f}{\partial x}(\xi_{x,\omega},\omega)(x-x_{0})\right|\leq g(\omega)|x-x_{0}|
+\]
+
+\end_inset
+
+Entonces 
+\begin_inset Formula $f_{x}$
+\end_inset
+
+ se diferencia de 
+\begin_inset Formula $f_{x_{0}}$
+\end_inset
+
+ en un múltiplo de una función integrable 
+\begin_inset Formula $g$
+\end_inset
+
+, por lo que también es integrable.
+ Para ver que la integral es derivable, basta tomar una sucesión 
+\begin_inset Formula $(x_{n})_{n}$
+\end_inset
+
+ de elementos de 
+\begin_inset Formula $X$
+\end_inset
+
+ que converge a un cierto 
+\begin_inset Formula $x$
+\end_inset
+
+ arbitrario y probar que existe 
+\begin_inset Formula $\lim_{n}\frac{F(x_{n})-F(x)}{x_{n}-x}$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $N\in\Sigma$
+\end_inset
+
+ de medida nula tal que para 
+\begin_inset Formula $\omega\notin N$
+\end_inset
+
+, 
+\begin_inset Formula $f^{\omega}$
+\end_inset
+
+ es derivable y por tanto 
+\begin_inset Formula 
+\[
+\frac{f(x_{n},\omega)-f(x,\omega)}{x_{n}-x}\rightarrow\frac{\partial f}{\partial x}(x,\omega)
+\]
+
+\end_inset
+
+Entonces 
+\begin_inset Formula $N\cup A$
+\end_inset
+
+ tiene medida nula y fuera de él, la sucesión
+\begin_inset Formula 
+\[
+\left(h_{n,\omega}:=\frac{f(x_{n},\omega)-f(x,\omega)}{x_{n}-x}\right)_{n}
+\]
+
+\end_inset
+
+ cumple que 
+\begin_inset Formula 
+\[
+|h_{n,\omega}|=\left|\frac{f(x_{n},\omega)-f(x,\omega)}{x_{n}-x}\right|=\left|\frac{\partial f}{\partial x}f(\eta_{x,x_{n},\omega},\omega)\right|\leq g(\omega)
+\]
+
+\end_inset
+
+ y, como además converge puntualmente, podemos aplicar el teorema de la
+ convergencia dominada y 
+\begin_inset Formula 
+\begin{eqnarray*}
+F'(x) & = & \lim_{n}\frac{F(x_{n})-F(x)}{x_{n}-x}=\lim_{n}\frac{\int f(x_{n},\omega)d\mu(\omega)-\int f(x_{n},\omega)d\mu(\omega)}{x_{n}-x}\\
+ & = & \lim_{n}\int\frac{f(x_{n},\omega)-f(x,\omega)}{x_{n}-x}d\mu(\omega)=\int\left(\lim_{n}\frac{f(x_{n},\omega)-f(x,\omega)}{x_{n}-x}\right)d\mu(\omega)\\
+ & = & \int\frac{\partial f}{\partial x}(x,\omega)d\mu(\omega)
+\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/fvv2/n4.lyx b/fvv2/n4.lyx
new file mode 100644
index 0000000..6628b45
--- /dev/null
+++ b/fvv2/n4.lyx
@@ -0,0 +1,688 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass book
+\use_default_options true
+\maintain_unincluded_children false
+\language spanish
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 0
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style french
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+\begin_inset Note Comment
+status open
+
+\begin_layout Section
+Espacios producto
+\end_layout
+
+\begin_layout Plain Layout
+Dados dos espacios medibles 
+\begin_inset Formula $(\Omega_{1},\Sigma_{1})$
+\end_inset
+
+ y 
+\begin_inset Formula $(\Omega_{2},\Sigma_{2})$
+\end_inset
+
+, llamamos 
+\series bold
+rectángulo medible
+\series default
+ en 
+\begin_inset Formula $\Omega:=\Omega_{1}\times\Omega_{2}$
+\end_inset
+
+ a los elementos de 
+\begin_inset Formula ${\cal R}:=\{A\times B\}_{A\in\Sigma_{1},B\in\Sigma_{2}}$
+\end_inset
+
+.
+ Llamamos 
+\series bold
+
+\begin_inset Formula $\sigma$
+\end_inset
+
+-álgebra producto
+\series default
+ de partes de 
+\begin_inset Formula $\Omega$
+\end_inset
+
+ a 
+\begin_inset Formula $\Sigma:=\Sigma_{1}\times\Sigma_{2}:=\sigma({\cal R})$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Plain Layout
+Como 
+\series bold
+teorema
+\series default
+, dados dos espacios de medida 
+\begin_inset Formula $(\Omega_{1},\Sigma_{1},\mu_{1})$
+\end_inset
+
+ y 
+\begin_inset Formula $(\Omega_{2},\Sigma_{2},\mu_{2})$
+\end_inset
+
+, existe una medida 
+\begin_inset Formula $\mu$
+\end_inset
+
+ en 
+\begin_inset Formula $\Sigma_{1}\times\Sigma_{2}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $\mu(A\times B)=\mu_{1}(A)\mu_{2}(B)$
+\end_inset
+
+ para cada 
+\begin_inset Formula $A\times B$
+\end_inset
+
+ medible.
+ Si además los dos espacios de medida son 
+\begin_inset Formula $\sigma$
+\end_inset
+
+-finitos, 
+\begin_inset Formula $\mu$
+\end_inset
+
+ es la única.
+ 
+\begin_inset Note Comment
+status open
+
+\begin_layout Plain Layout
+
+\series bold
+Demostración:
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Cambio de variable
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sea 
+\begin_inset Formula $(\Omega,\Sigma,\mu)$
+\end_inset
+
+ un espacio de medida, 
+\begin_inset Formula $(\Omega',\Sigma')$
+\end_inset
+
+ un espacio medible y 
+\begin_inset Formula $g:\Omega\rightarrow\Omega'$
+\end_inset
+
+ medible, llamamos 
+\series bold
+medida imagen
+\series default
+ de 
+\begin_inset Formula $\mu$
+\end_inset
+
+ a través de 
+\begin_inset Formula $g$
+\end_inset
+
+ a la medida 
+\begin_inset Formula $\nu:=\mu g^{-1}:\Sigma'\rightarrow[0,+\infty]$
+\end_inset
+
+ dada por 
+\begin_inset Formula $\nu(A):=\mu(g^{-1}(A))$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Una función 
+\begin_inset Formula $\Sigma'$
+\end_inset
+
+-medible 
+\begin_inset Formula $f:\Omega'\rightarrow\mathbb{C}$
+\end_inset
+
+ es 
+\begin_inset Formula $\nu$
+\end_inset
+
+-integrable si y sólo si 
+\begin_inset Formula $f\circ g$
+\end_inset
+
+ es 
+\begin_inset Formula $\mu$
+\end_inset
+
+-integrable, y si 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\begin_inset Formula $\nu$
+\end_inset
+
+-integrable o 
+\begin_inset Formula $f\geq0$
+\end_inset
+
+ se tiene 
+\begin_inset Formula 
+\[
+\int f\,d\nu=\int f\circ g\,d\mu
+\]
+
+\end_inset
+
+
+\series bold
+Demostración:
+\series default
+ Para 
+\begin_inset Formula $A\in\Sigma'$
+\end_inset
+
+ y 
+\begin_inset Formula $x\in\Omega$
+\end_inset
+
+, 
+\begin_inset Formula $\chi_{g^{-1}(A)}(x)=\chi_{A}(g(x))=(\chi_{A}\circ g)(x)$
+\end_inset
+
+, luego 
+\begin_inset Formula $\chi_{g^{-1}(A)}=\chi_{A}\circ g$
+\end_inset
+
+ y 
+\begin_inset Formula $\int_{\Omega'}\chi_{A}d\nu=\nu(A)=\mu(g^{-1}(A))=\int_{\Omega}\chi_{g^{-1}(A)}d\mu=\int_{\Omega}(\chi_{A}\circ g)d\mu$
+\end_inset
+
+.
+ Con esto, una función simple 
+\begin_inset Formula $f:\Sigma'\rightarrow[0,+\infty]$
+\end_inset
+
+ es 
+\begin_inset Formula $\nu$
+\end_inset
+
+-integrable si y sólo si 
+\begin_inset Formula $f\circ g$
+\end_inset
+
+ es 
+\begin_inset Formula $\mu$
+\end_inset
+
+-integrable, y entonces 
+\begin_inset Formula $\int f\,d\nu=\int f\circ g\,d\mu$
+\end_inset
+
+.
+ Usando el teorema de la convergencia monótona podemos extender este resultado
+ a funciones medibles positivas, y el de la convergencia dominada nos da
+ el resultado para el caso general.
+\end_layout
+
+\begin_layout Standard
+Dada una función 
+\begin_inset Formula $\alpha:[a,b]\rightarrow\mathbb{R}$
+\end_inset
+
+ creciente y continua por la derecha, llamamos 
+\series bold
+medida de Lebesgue-Stieltjes
+\series default
+ asociada a 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ o 
+\series bold
+medida de Borel inducida
+\series default
+ a la única medida de Borel con 
+\begin_inset Formula $\mu_{\alpha}((a,b])=\alpha(b)-\alpha(a)$
+\end_inset
+
+, y que se construye de forma similar a la de Lebesgue.
+\end_layout
+
+\begin_layout Standard
+Como 
+\series bold
+teorema
+\series default
+, si 
+\begin_inset Formula $f:[a,b]\rightarrow\mathbb{R}$
+\end_inset
+
+ es acotada y 
+\begin_inset Formula $D(f):=\{x\in[a,b]:f\text{ es discontinua en }x\}$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable Riemann-Stieltjes con respecto a 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ en 
+\begin_inset Formula $[a,b]$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $\mu_{\alpha}(D(f))=0$
+\end_inset
+
+, y en tal caso 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\begin_inset Formula $\mu_{\alpha}$
+\end_inset
+
+-integrable y 
+\begin_inset Formula $\int_{[a,b]}f\,d\mu_{\alpha}=\int_{a}^{b}f\,d\alpha$
+\end_inset
+
+.
+ La demostración de esto es similar a la correspondiente para integrales
+ de Riemann simples.
+\end_layout
+
+\begin_layout Standard
+Dado un espacio de medida 
+\begin_inset Formula $(\Omega,\Sigma,\mu)$
+\end_inset
+
+ finito, llamamos 
+\series bold
+variables aleatorias
+\series default
+ a las funciones medibles 
+\begin_inset Formula $f:\Omega\rightarrow\mathbb{R}$
+\end_inset
+
+, y tenemos que 
+\begin_inset Formula $\mu f^{-1}$
+\end_inset
+
+ es una medida finita en 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ tal que 
+\begin_inset Formula $f=f\circ id$
+\end_inset
+
+ es 
+\begin_inset Formula $\mu$
+\end_inset
+
+-integrable si y sólo si 
+\begin_inset Formula $id$
+\end_inset
+
+ es 
+\begin_inset Formula $\mu f^{-1}$
+\end_inset
+
+-integrable, y entonces 
+\begin_inset Formula $\int_{\Omega}f\,d\mu=\int_{\mathbb{R}}id\,d\mu f^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\series bold
+función de distribución
+\series default
+ de 
+\begin_inset Formula $f$
+\end_inset
+
+ a 
+\begin_inset Formula $F(x):=\mu(\{f\leq x\})$
+\end_inset
+
+ o a 
+\begin_inset Formula $\varphi(x):=\mu(\{f>x\})=\mu(\Omega)-F(x)$
+\end_inset
+
+.
+ Sea 
+\begin_inset Formula $f:\Omega\rightarrow(a,b]$
+\end_inset
+
+ una variable aleatoria, 
+\begin_inset Formula $F(x):=\mu(\{f\leq x\})$
+\end_inset
+
+ y 
+\begin_inset Formula $\varphi(x):=\mu(\{f>x\})$
+\end_inset
+
+, entonces 
+\begin_inset Formula $f$
+\end_inset
+
+ es integrable y 
+\begin_inset Formula 
+\[
+\int f\,d\mu=\int_{[a,b]}id\,d\mu f^{-1}=\int_{a}^{b}id\,dF=-\int_{a}^{b}id\,d\varphi
+\]
+
+\end_inset
+
+
+\series bold
+
+\begin_inset Newpage clearpage
+\end_inset
+
+Demostración: 
+\series default
+La identidad es continua y por tanto integrable Riemann-Stieltjes respecto
+ a 
+\begin_inset Formula $F$
+\end_inset
+
+ o 
+\begin_inset Formula $\varphi$
+\end_inset
+
+, y como 
+\begin_inset Formula $F(y)-F(x)=-(\varphi(y)-\varphi(x))$
+\end_inset
+
+, se tiene 
+\begin_inset Formula $\int_{a}^{b}id\,dF=-\int_{a}^{b}id\,d\varphi$
+\end_inset
+
+, pero entonces 
+\begin_inset Formula $id$
+\end_inset
+
+ es 
+\begin_inset Formula $\mu f^{-1}$
+\end_inset
+
+-integrable.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $f$
+\end_inset
+
+ no es acotada, podemos aplicar este resultado a los conjuntos 
+\begin_inset Formula $E_{a,b}:=\{a<f\leq b\}$
+\end_inset
+
+ para 
+\begin_inset Formula $a<b$
+\end_inset
+
+ cualesquiera definiendo 
+\begin_inset Formula $\mu_{a,b}(E):=\mu(E\cap E_{a,b})$
+\end_inset
+
+ y usando esta medida.
+\end_layout
+
+\begin_layout Standard
+La restricción de que 
+\begin_inset Formula $f$
+\end_inset
+
+ sea acotada se puede suprimir definiendo 
+\begin_inset Formula 
+\[
+\int_{-\infty}^{+\infty}id\,d\varphi:=\lim_{\begin{subarray}{c}
+a\to-\infty\\
+b\to+\infty
+\end{subarray}}\int_{a}^{b}id\,d\varphi
+\]
+
+\end_inset
+
+ cuando el límite existe.
+ En tal caso se cumple
+\begin_inset Formula 
+\begin{eqnarray*}
+\int_{\Omega}f^{+}d\mu=\lim_{b\rightarrow+\infty}\int_{E_{0,b}}f\,d\mu<+\infty & \text{ y } & \int_{\Omega}f^{-}d\mu=-\lim_{a\rightarrow-\infty}\int_{E_{a,0}}f\,d\mu<+\infty
+\end{eqnarray*}
+
+\end_inset
+
+por lo que 
+\begin_inset Formula $f$
+\end_inset
+
+ es 
+\begin_inset Formula $\mu$
+\end_inset
+
+-integrable.
+\end_layout
+
+\begin_layout Standard
+De aquí que, si 
+\begin_inset Formula $f:\Omega\rightarrow\mathbb{R}$
+\end_inset
+
+ es medible, entonces 
+\begin_inset Formula $f\in{\cal L}^{1}(\Omega)$
+\end_inset
+
+ si y sólo si 
+\begin_inset Formula $\int_{-\infty}^{+\infty}id\,d\varphi$
+\end_inset
+
+ es finita, y en tal caso 
+\begin_inset Formula $\int_{\Omega}f\,d\mu=-\int_{-\infty}^{+\infty}id\,d\varphi$
+\end_inset
+
+.
+ Entonces:
+\end_layout
+
+\begin_layout Enumerate
+Si 
+\begin_inset Formula $f:\Omega\rightarrow[a,b]$
+\end_inset
+
+ es medible con 
+\begin_inset Formula $\varphi(x):=\mu(\{f>x\})$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi:[a,b]\rightarrow\mathbb{R}$
+\end_inset
+
+ es continua, entonces 
+\begin_inset Formula $\phi\circ f$
+\end_inset
+
+ es integrable y
+\begin_inset Formula 
+\[
+\int_{\Omega}\phi\circ f\,d\mu=-\int_{a}^{b}\phi\,d\varphi
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si es 
+\begin_inset Formula $f:\Omega\rightarrow\mathbb{R}$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi:\mathbb{R}\rightarrow[0,+\infty)$
+\end_inset
+
+, entonces 
+\begin_inset Formula 
+\[
+\int_{\Omega}\phi\circ f\,d\mu=-\int_{-\infty}^{+\infty}\phi\,d\varphi
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Si es 
+\begin_inset Formula $\phi:\mathbb{R}\rightarrow\mathbb{R}$
+\end_inset
+
+ y 
+\begin_inset Formula $\phi\circ f$
+\end_inset
+
+ es integrable, esta igualdad también se cumple.
+\end_layout
+
+\begin_layout Standard
+Llamamos 
+\begin_inset Formula ${\cal L}_{\phi}(\mu)$
+\end_inset
+
+ al conjunto de funciones medibles 
+\begin_inset Formula $f$
+\end_inset
+
+ tales que 
+\begin_inset Formula $\phi\circ f$
+\end_inset
+
+ es integrable, y para 
+\begin_inset Formula $\phi(x)=|x|^{p}$
+\end_inset
+
+, escribimos 
+\begin_inset Formula ${\cal L}^{p}(\mu):={\cal L}_{\phi}(\mu)$
+\end_inset
+
+.
+ Esto es compatible con la definición inicial de 
+\begin_inset Formula ${\cal L}^{1}(\mu)$
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document