From 6fcc9b8e040f6314edaaee75c895cc6fc4fc38d6 Mon Sep 17 00:00:00 2001 From: Robert Date: Wed, 31 Mar 2021 19:14:34 +0200 Subject: [PATCH] Finished product measure --- chapters/measures_integrals.tex | 1 + chapters/sections/product_measures.tex | 311 +++++++++++++++++++++++++ script.pdf | Bin 1061516 -> 1081504 bytes script.tex | 1 + 4 files changed, 313 insertions(+) create mode 100644 chapters/sections/product_measures.tex diff --git a/chapters/measures_integrals.tex b/chapters/measures_integrals.tex index d25da0b..1f7f583 100644 --- a/chapters/measures_integrals.tex +++ b/chapters/measures_integrals.tex @@ -9,4 +9,5 @@ \subfile{sections/contents_measures.tex} \subfile{sections/integrals.tex} \subfile{sections/int_real_nums.tex} + \subfile{sections/product_measures.tex} \end{document} \ No newline at end of file diff --git a/chapters/sections/product_measures.tex b/chapters/sections/product_measures.tex new file mode 100644 index 0000000..a7872a3 --- /dev/null +++ b/chapters/sections/product_measures.tex @@ -0,0 +1,311 @@ +% !TeX root = ../../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} +\section{Product measures and Fubini's Theorem} + +\begin{eg} + Let + \[ + f: [0, 1] \times [0, 1] \longrightarrow [0, \infty) + \] + Question: What is the volume (or the $\lambda^2$ measure) under the graph of $f$, i.e. + \[ + \set[0 \le z \le f(x, y)]{(x, y, z) \in \realn^3} + \] + The possibilities are + \begin{gather*} + \int f \dd\lambda^2 \\ + \int_0^1 \int_0^1 f(x, y) \dd x \dd y ~~\text{ or }~~ \int_0^1 \int_0^1 f(x, y) \dd y \dd x + \end{gather*} +\end{eg} +From now on we define $\measure$ and $(\Phi, \setfamb, \nu)$ to be measure spaces. +\begin{defi} + The product $\sigma$-algebra $\setfam \otimes \setfamb$ is the smallest $\sigma$-algebra on $\Omega \times \Phi$ that contains + all sets of type $A \times B$ for $A \in \setfam, B \in \setfamb$. + + Examples for $A \times B$ are + + \begin{center} + \begin{minipage}{.49\linewidth} + \begin{center} + \begin{tikzpicture} + \draw[very thick, ->] (0, 0) -- (5, 0); + \draw[very thick, ->] (0, 0) -- (0, 5); + + \draw (1, 1) -- (4, 1) -- (4, 4) -- (1, 4) -- (1, 1); + + \draw[dashed] (0, 4) -- (1, 4); + \draw[dashed] (0, 1) -- (1, 1); + \draw[dashed] (1, 0) -- (1, 1); + \draw[dashed] (4, 0) -- (4, 1); + + \node[below] at (2.5, 0) {$A$}; + \node[left] at (0, 2.5) {$B$}; + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{.49\linewidth} + \begin{center} + \begin{tikzpicture} + \draw[very thick, ->] (0, 0) -- (5, 0); + \draw[very thick, ->] (0, 0) -- (0, 5); + + \draw (1, 1) -- (2, 1) -- (2, 2) -- (1, 2) -- (1, 1); + \draw (3, 1) -- (4, 1) -- (4, 2) -- (3, 2) -- (3, 1); + \draw (1, 3) -- (2, 3) -- (2, 4) -- (1, 4) -- (1, 3); + \draw (3, 3) -- (4, 3) -- (4, 4) -- (3, 4) -- (3, 3); + + \draw[dashed] (0, 4) -- (1, 4); + \draw[dashed] (0, 3) -- (1, 3); + \draw[dashed] (0, 2) -- (1, 2); + \draw[dashed] (0, 1) -- (1, 1); + \draw[dashed] (4, 0) -- (4, 1); + \draw[dashed] (3, 0) -- (3, 1); + \draw[dashed] (2, 0) -- (2, 1); + \draw[dashed] (1, 0) -- (1, 1); + + \node[left] at (0, 3.5) {$B$}; + \node[left] at (0, 1.5) {$B$}; + \node[below] at (1.5, 0) {$A$}; + \node[below] at (3.5, 0) {$A$}; + \end{tikzpicture} + \end{center} + \end{minipage} + \end{center} + \pagebreak NO examples for $A \times B$ are + + \begin{center} + \begin{minipage}{.49\linewidth} + \begin{center} + \begin{tikzpicture} + \draw[very thick, ->] (0, 0) -- (5, 0); + \draw[very thick, ->] (0, 0) -- (0, 5); + + \draw (1, 1) -- (2, 1) -- (2, 2) -- (1, 2) -- (1, 1); + \draw (3, 1) -- (4, 1) -- (4, 2) -- (3, 2) -- (3, 1); + \draw (1, 3) -- (2, 3) -- (2, 4) -- (1, 4) -- (1, 3); + + \draw[dashed] (0, 4) -- (1, 4); + \draw[dashed] (0, 3) -- (1, 3); + \draw[dashed] (0, 2) -- (1, 2); + \draw[dashed] (0, 1) -- (1, 1); + \draw[dashed] (4, 0) -- (4, 1); + \draw[dashed] (3, 0) -- (3, 1); + \draw[dashed] (2, 0) -- (2, 1); + \draw[dashed] (1, 0) -- (1, 1); + + \node[left] at (0, 3.5) {$B$}; + \node[left] at (0, 1.5) {$B$}; + \node[below] at (1.5, 0) {$A$}; + \node[below] at (3.5, 0) {$A$}; + \end{tikzpicture} + \end{center} + \end{minipage} + \begin{minipage}{.49\linewidth} + \begin{center} + \begin{tikzpicture} + \draw[very thick, ->] (0, 0) -- (5, 0); + \draw[very thick, ->] (0, 0) -- (0, 5); + + \draw (2.5, 2.5) circle[radius=1.5]; + + \node[below] at (0, 0) {\quad}; + \end{tikzpicture} + \end{center} + \end{minipage} + \end{center} + A measure $\vartheta$ defined on $\setfam \otimes \setfamb$ is said to be a product measure of $\mu, \nu$ if + \[ + \vartheta(A \times B) = \mu(A) \nu(B) ~~A \in \setfam, B \in \setfamb + \] +\end{defi} + +\begin{rem} + Product measures always exist. For $\sigma$-finite measure spaces they are unique. Notation: + \[ + \mu \otimes \nu ~~\text{ or }~~ \mu^2 = \mu \otimes \mu + \] +\end{rem} + +\begin{eg} + $\realn$ with Lebesgue measure $\lambda$. $\lambda^2$ is the product measure on $\realn^2$. + \begin{align*} + \lambda^2([a, b] \times [c, d]) &= \lambda([a, b])\lambda([c, d]) \\ + &= (b - a)(d - c) + \end{align*} + This means the product measure characterizes the area. Analogously this can be extended for $\lambda^3$, $\lambda^4$ etc. +\end{eg} + +\begin{eg} + Consider + \begin{align*} + f: \realn^2 &\longrightarrow \realn \\ + f &= \sum_{n=0}^{\infty} \left(\charfun_{[n, n+1)^2} - \charfun_{[n+1, n+2) \times [n, n+1)}\right) + \end{align*} + + \begin{center} + \begin{tikzpicture} + \draw[->, very thick] (0, 0) -- (0, 3.5) node[above] {$y$}; + \draw[->, very thick] (0, 0) -- (5.5, 0) node[right] {$x$}; + + \foreach \x in {1,...,5} \draw (\x, -0.1) -- (\x, 0.1); + \foreach \y in {1,...,3} \draw (-0.1, \y) -- (0.1, \y); + + \draw (0, 1) -- (3, 1); + \draw (1, 2) -- (4, 2); + \draw (2, 3) -- (4, 3); + \draw (1, 2) -- (1, 0); + \draw (2, 3) -- (2, 0); + \draw (3, 3) -- (3, 1); + \draw (4, 3) -- (4, 2); + + \node at (4.5, 2.5) {$\cdots$}; + \node at (0.5, 0.5) {$1$}; + \node at (1.5, 0.5) {$-1$}; + \node at (1.5, 1.5) {$1$}; + \node at (2.5, 1.5) {$-1$}; + \node at (2.5, 2.5) {$1$}; + \node at (3.5, 2.5) {$-1$}; + + \draw[dashed] (0, 0.8) -- (5, 0.8) node[right] {$=0$}; + \draw[dashed] (0.4, 0) -- (0.4, 3) node[above] {$=1$}; + \draw[dashed] (1.8, 0) -- (1.8, 3) node[above] {$=0$}; + \end{tikzpicture} + \end{center} + \begin{align*} + \iint f(x, y) \dd x \dd y = 0 && \iint f(x, y) \dd y \dd x = 1 + \end{align*} + But the integral $\int f \dd\lambda^2$ doesn't exist + \[ + \int \abs{f} \dd\lambda^2 = \sum_{n=0}^{\infty} 2 = \infty + \] +\end{eg} + +\begin{thm}[Tonelli's Theorem] + Let $f: \Omega \times \Phi \rightarrow [0, \infty)$ be measurable (in terms of $\setfam \otimes \setfamb$). + Then the functions + \[ + \omega \longmapsto f(\omega, \phi) + \] + are measurable for almost all $\phi \in \Phi$. Analogously + \[ + \phi \longmapsto f(\omega, \phi) + \] + is measurable for almost all $\omega \in \Omega$. + \begin{align*} + \phi &\longmapsto \int f(\omega, \phi) \dd\mu(\omega) \text{ measurable} \\ + \omega &\longmapsto \int f(\omega, \phi) \dd\mu(\phi) \text{ measurable} + \end{align*} + and + \begin{align*} + \int f(\omega, \phi) \dd (\mu \otimes \nu) (\omega, \phi) &= \iint f(\omega, \phi) \dd\mu(\omega) \dd\nu(\phi) \\ + &= \iint f(\omega, \phi) \dd\nu(\phi) \dd\mu(\omega) + \end{align*} + Furthermore, $f$ is integrable in terms of $\mu \otimes \nu$ is one of the above integrals is finite. +\end{thm} +\begin{proof} + Without proof. +\end{proof} + +\begin{cor}[Cavalieri's Principle] + Let $A \subset \setfam \otimes \setfamb$. Define + \[ + A_{\omega} = \set[(\omega, \phi) \in A]{\phi \in \Phi} + \] + Then + \[ + \omega \longmapsto \nu(A_{\omega}) + \] + is measurable and + \[ + (\mu \otimes \nu)(A) = \int \nu(A_{\omega}) \dd\mu(A_{\omega}) + \] +\end{cor} +\begin{proof} + It is easy to see + \begin{equation} + (\omega, \phi) \in A \iff \phi \in A_{\omega} + \end{equation} + Thus we can see + \begin{equation} + \charfun_A(\omega, \phi) = \charfun_{A_{\omega}} (\phi) + \end{equation} + And then + \begin{equation} + \begin{split} + (\mu \otimes \nu)(A) &= \int \charfun_A(\omega, \phi) \dd{(\mu \otimes \nu)}(\omega, \phi) \\ + &= \iint \underbrace{\charfun_A (\omega, \phi)}_{\charfun_{A_{\omega}}(\phi)} \dd{\nu}(\phi) \dd{\mu}(\omega) \\ + &= \int \nu(A_{\omega}) \dd{\mu}(\omega) + \end{split} + \end{equation} +\end{proof} + +\begin{thm}[Fubini's Theorem] + Let $f: \Omega \times \Phi \rightarrow \field$ be measurable with measures $\mu, \nu$, which is integrable in terms of $\mu \otimes \nu$. + Then the functions $\omega \mapsto f(\omega, \phi)$ are measurable and integrable for $\nu$-almost every $\phi \in \Phi$, + and the functions $\phi \mapsto f(\omega, \phi)$ are measurable and integrable for $\mu$-almost every $\omega \in \Omega$. + The functions + \begin{align*} + \omega &\longmapsto \int f(\omega, \phi) \dd{\nu}(\phi) && \phi \longmapsto \int f(\omega, \phi) \dd{\mu}(\omega) + \end{align*} + are measurable and integrable, and + \begin{align*} + \int f(\omega, \phi) \dd{(\mu \otimes \nu)} &= \iint f(\omega, \phi) \dd{\nu}(\phi) \dd{\mu}(\omega) \\ + &= \iint f(\omega, \phi) \dd{\mu}(\omega) \dd{\nu}(\phi) + \end{align*} +\end{thm} +\begin{proof} + Without proof. +\end{proof} + +\begin{cor} + Let $a_i, b_i \in \realn$, $a_i < b_i ~~\forall i \in \set{1, \cdots, n}$. + \[ + D = [a_1, b_1] \times [a_2, b_2] \times \cdots \times [a_n, b_n] + \] + Let $f: D \rightarrow \realn$ be continuous or bounded. Then + \[ + \int_D f \dd{\lambda^n} = \int\limits_{a_1}^{b_1} \cdots \int\limits_{a_n}^{b_n} f(x_1, \cdots, x_n) \dd{x_n} \cdots \dd{x_1} + \] + and the order of integration is irrelevant. +\end{cor} +\begin{proof} + $f$ is bounded by $k \in \realn$ (continuous with compact domain) + \begin{equation} + \int_D \abs{f} \dd{\lambda^n} \le \int_D k \dd{\lambda^n} = k \cdot (b_1 - a_1)(b_2 - a_2) \cdots (b_n - a_n) < \infty + \end{equation} + $f$ is $\lambda^n$-integrable. By applying Fubini's theorem the desired statement follows. +\end{proof} + +\begin{eg} + Calculate the center of mass of + \[ + A = \set[x \ge y^2 \wedge x \le 1]{(x, y) \in \realn^2} + \] + + \begin{center} + \begin{tikzpicture}[scale=2] + \draw[->, thick] (-0.5, 0) -- (3.5, 0); + \draw[->, thick] (0, -1.2) -- (0, 1.2); + + \draw[domain=-1:1, smooth, variable=\y] plot({3*\y*\y}, {\y}); + \draw[dashed] (3, -1) -- (3, 1); + \end{tikzpicture} + \end{center} + The center of mass is defined by + \[ + \int \begin{pmatrix} + x \\ y + \end{pmatrix} \underbrace{\dd{\lambda^2}(x, y)}_{\dd A} + \] + In component form this is + \begin{align*} + \int_A x \dd{\lambda^2}(x, y) &= \int x \charfun_A(x, y) \dd{\lambda^2}(x, y) \\ + &= \int_{[0, 1] \times [-1, 1]} x \charfun_A(x, y) \dd{\lambda^2}(x, y) \\ + &= \int_0^1 \int_{-1}^1 x \charfun_{[-\sqrt{x}, \sqrt{x}]}(y) \dd{y}\dd{x} \\ + &= \int_0^1 x \cdot 2 \cdot \sqrt{x} \dd{x} = \frac{4}{5} + \end{align*} + Meaning the center of mass is at $(\frac{4}{5}, 0)$. +\end{eg} +\end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index feb5eaa5121fd2678dc6006a4f5a2b3f80f78b8d..ce95a2c7021e0ea7fce56b5356f42ada2e8c459e 100644 GIT binary patch delta 289938 zcmce9cYIXE`uAtg)`TP^+1+FlT0(~;klk#B&{RSPrK(qj04mb;ib@mpT0oTIQ3h#( zNEfB=RX~c@O20O`aIJt1v4B|K?=y2YIde7}x!&LB{R6Vk%qhE1Z!`16XYCh1c-6Y~ z*|Wb($_TaaH_L0=Hu=`kL&pprp6o4AJJfWm10D5JZz?V>zNy%cf5pY4)Zg6=)IGKa z>SR}Pufa5Q2mLp?;!_j)^mqC%c28ddhS7hw(tl&%S4}aKYi)_;L0H6FXGfR^R0>q3e=c_P+J+W5(SzU9dA=}!u=-+)Tm`>i{T6cp^Y*j4 zHriXRGn>L}eH%-CXIR6N4ukB_E!L*#F$BgirgkiOR_)L+#}T40)I0BH^_t*7wXMIS zdM4OM{m|b@{YR*eIyTTrtrPC8CIvgIQT$yREK_?$daJENoz*k=`$4Fa`e3WR>ez6Z zn%la&szkc0n_Bl(pO2KOJxjYkRq9YnTa~GwmG)NOZ`Dbi(x$gMs&!{Ir){^VN_(m6 z+ICatm3C9Rw(FyY+jLStY1dmlgufHpcUOmE&fTGpsg7(URzh`HeQ=bMP?}>30WC_`JU>tu=F9~q z1Z#N1L>}TufmC+reS0D6@R8YR2?Tt3>gkEiru5r0>`6ymw<)=XVVqzt?Sj&+?ZY))$(sTVJR3fS(uredbK>VI}2c;ppja?0c$t z`{H-+?AiW&&Uf9Hlpc7nZ%+NE9~^A1o7bbQ``fH=*^?LcJ@v&~OPV=TuIv7*(yZB4 zFLwX;g5-;@z1;dj(d^n&=6v|{xf`1_>r?yWGvhw^=(kV0DBsP>>==3Q;c4nWE)1S_ zP4~NIWj^}wnns^q-LOm2a@WZweP0>5u-ORq#4CIEz1h-JnEA$_V~4SaUy+~QH= zK520Lj}d+5?|*s9f{#9%(y2@H;@952d(O@2XTr196JuZB`FiKp&(D4|dfV3bg7wF3 zat?D&QEqPe@B_=fo%Z^wmbtT9GH1`dh+y?qV2VRS$?+rG0UR1 zU%tG3bn{MozRP*!^(md2?Vk2~?w}94uPc3Md(r$2-@Ct6URZYNrO=tN18#HN*Zz*4KEOPrt<)ePQxGi<|Zzl`J zEPP?~`#=5roAtleDDB~0`f2yu%C6oTU3>MiQ_Ej|zFC*AYqDQ2eo*t|=AUK`9CFK> z?+sq@w>3X!*MH&H+(8%4tQfvvVD|Lq%33C`RS)%bvDr`98)R~(lGK;FcvX|zDzsNj z`0~6VA>rPT5N>ZsQ(s{rjoz>}AJ*o>v3X6cMTA=OM!39l^`^e$tF5*#i>UvBxU6g2 zTYaFcv%0PWDJuwPr_SozU3#mV@ON<6K5DPdozzy{NOan(r#tsh-@7JLZQo_8de`Ul zw}V=%YnghvYbx8{+HT_mo4a;W7xbvXj=yDUrw;1i$O?F2U%YuGL0=1h(3=L&)US~Wp8y)tHb3#!q$5n1?nw5GT3>K!^P6~m_4Ql%h=%P>XbBrCj93& zI0~4jS6ZsN?RCe-NYB+alaDQ)W~ra$EB4X1(!bDGT+CnCP-xo9U+F6*1*22>zG7(U ztxEF@qM0<#?#x%Xg7T0XRG`ArLyi{o3FarS1jvVypI&GLfZ*ZancRG;VC;iKS8N)_W6A5!9xxwCzX|x zN}Vyd4(Mv~;D(O|XR@*R3b(r+)jdQ&s&#O#+HXiYpHi~2j^C!iIeatp->=~5&xS?*y(;I=fH?i22Wx_DGwQ-B@Hu@tCV@MSlgkFoU`7+?Qn~u&T$fXV#e=>9v_{XZsc+? 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