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+#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