From f4449040215ccfb66cd871f8e404d361facf8a0d Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Fri, 19 Oct 2012 21:26:22 +0200 Subject: [PATCH] Nachtrag vom 19.10.2012 --- documents/Analysis III/Analysis-III.pdf | Bin 753830 -> 759348 bytes documents/Analysis III/Analysis-III.tex | 295 ++++++++++++++++-------- documents/Analysis III/mathe.sty | 4 +- 3 files changed, 197 insertions(+), 102 deletions(-) diff --git a/documents/Analysis III/Analysis-III.pdf b/documents/Analysis III/Analysis-III.pdf index a276504e07446c2291e2c5d1f8763eea8892a3ed..355260984eb9e749ddc9a4c5d429699fec386412 100644 GIT binary patch delta 149783 zcmcG#Q*>rQv@IOlw#_e2I<{@A!;abE7uy}%wr$%+$F}|VIrrRi#vS9okN2gj=BgT_ z_I%j8)||C=u|HE|B@-H@l9&V|3lj$%^LB3 zni^$#pQVbW>L|TdA$=eMR}WRO{$F`$!{@tD`Pz#plfce04clT5E!6XV`TzgA5Q>yh!z>k=$_1>NG=IN`}^bh%2xjzXaC$1 zB#~Z^fk9_`A$LA^d_sg}TR4IOOi6q{m2sxzgzE!p7vt4JXLYuXN9|h4J6Q#mAaf6~ zPjoMH%((4=y4nUOr92u_uvgy5&vBu^^(KPlm?! zve!Atz^KN?FmQPLZnZXqJZMG^4W3hQwOpr2;HUkE6KHbAUf%T65Vxw@uRc^WHr1iP 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wide} + {\pgfqpoint{-1pt}{-1pt}}% + {\pgfqpoint{7pt}{7pt}}% + {\pgfqpoint{6pt}{6pt}}% + { + \pgfsetlinewidth{0.7pt} + \pgfpathmoveto{\pgfqpoint{0pt}{6pt}} + \pgfpathlineto{\pgfqpoint{6.1pt}{-0.1pt}} + \pgfusepath{stroke} + } + + % Draw two intersecting lines + \draw[thick, red] (-1,-1) coordinate (a) -- (2,-1) coordinate (b); + \draw[thick, green] ( 1,-1) coordinate (c) -- (1, 2) coordinate (d); + + \fill[pattern=north east lines wide, pattern color=red!50] (a) -- (b) -- (2,2) -- (-1,2) -- (a); + \fill[pattern=north west lines wide, pattern color=green!50] (a) -- (1,-1) -- (1,2) -- (-1,2) -- (a); + + \draw[thick, green] (c) -- (d); + \draw[thick, red] (a) -- (b); + + + % Draw axes + \draw [<->,thick] (0,2.5) node (yaxis) [above] {$x_2$} + |- (2.5,0) node (xaxis) [right] {$x_1$}; + \node[red] at (1.5,2.8) {$H_2^+(-1)$}; + \node[green] at (1.5,2.3) {$H_1^-(1)$}; +\end{tikzpicture} \begin{satz}[Erzeuger der Borelschen $\sigma$-Algebra auf $\mdr^d$] \label{Satz 1.4} Es seien $\ce_1,\ce_2,\ce_3$ wie folgt definiert: \begin{align*} -\ce_1&:=\{(a,b):a,b\in\mdq^d,a\le b\}\\ -\ce_2&:=\{(a,b]:a,b\in\mdq^d, a\le b\}\\ -\ce_3&:=\{H^-_k(\alpha):\alpha\in\mdq, k=1,\dots,d\} +\ce_1&:=\Set{(a,b) | a,b\in\mdq^d,a\le b}\\ +\ce_2&:=\Set{(a,b] | a,b\in\mdq^d, a\le b}\\ +\ce_3&:=\Set{H^-_k(\alpha) | \alpha\in\mdq, k \in \Set{1,\dots,d}} \end{align*} Dann gilt: \[\fb_d=\sigma(\ce_1)=\sigma(\ce_2)=\sigma(\ce_3)\] @@ -383,96 +424,129 @@ Entsprechendes gilt für die anderen Typen von Intervallen und Halbräumen. \end{satz} \begin{beweis} +\[\fb_d + \stackrel{(1)}{\subseteq} \sigma(\ce_1) + \stackrel{(2)}{\subseteq} \sigma(\ce_2) + \stackrel{(3)}{\subseteq} \sigma(\ce_3) + \stackrel{(4)}{\subseteq} \fb_d +\] \begin{enumerate} - \item Sei $G\in\co(\mdr^d), \fm:=\{(a,b):a,b\in\mdq^d,a\le b, (a,b)\subseteq G\}$. - Dann ist $\fm$ abzählbar und $G=\bigcup_{I\in\fm}I$. Also - gilt: - \[G\in\sigma(\ce_1)\implies \fb_d=\sigma(\co(\mdr^d))\subseteq\sigma(\ce_1)\] - \item Sei $(a,b)\in\ce_1$.\\ + \item Sei $G\in\co(\mdr^d), \fm:=\Set{(a,b) | a,b \in \mdq^d, \; a\le b, \; (a,b)\subseteq G}$.\\ + Dann ist $\fm$ abzählbar und $G=\bigcup_{I\in\fm}I$.\\ + Also gilt: + \[\co(\mdr^d) \subseteq \sigma(\ce_1)\] + \[G\in\sigma(\ce_1)\implies \fb_d=\sigma(\co(\mdr^d))\stackrel{1.3}{\subseteq}\sigma(\ce_1)\] + \item Sei $a=(a_1, \dots,a_d), b=(b_1,\dots,b_d) \in \mdq^d$ und $a \leq b$ sowie $(a, b)\in\ce_1$.\\ \textbf{Fall 1:} $(a,b)=\emptyset\in\ce_2\subseteq\sigma(\ce_2)$\\ - \textbf{Fall 2:} $(a,b)\ne\emptyset, a=(a_1\dots,a_d), b=(b_1\dots,b_d)$.\\ + \textbf{Fall 2:} $(a,b)\ne\emptyset$.\\ Dann gilt für alle $j\in\{1,\dots,d\}:a_j0\exists n_c\in\mdn:\forall n\ge n_c: x_n> c\] -\item Sei $(a_n)$ eine Folge in $[0,+\infty]$. Es gilt -\[\sum_{n=1}^\infty a_n=\sum a_n = +\infty\] -genau dann wenn $a_j=+\infty$ für ein $j\in\mdn$ oder, falls alle $a_j<+\infty$, wenn $\sum a_n$ divergiert. + \item Sei $(x_n)$ eine Folge in $[0,+\infty]$. Es gilt: + \[x_n\stackrel{n\to\infty}{\to}\infty:\iff \forall c>0\;\exists n_c\in\mdn:\forall n\ge n_c: x_n> c\] + \item Sei $(a_n)$ eine Folge in $[0,+\infty]$. Es gilt + \[\sum_{n=1}^\infty a_n=\sum a_n = +\infty :\Leftrightarrow + \begin{cases} + \exists n \in \mdn \text{ mit } a_n = +\infty \text{ oder }\\ + \sum a_n \text{ divergiert} + \end{cases} + \] \end{enumerate} -Wegen 13.1 Ana I können Reihen der obigen Form beliebig umgeordnet werden, ohne dass sich ihr Wert verändert. +Wegen Ana I, 13.1 können Reihen der obigen Form beliebig umgeordnet +werden, ohne dass sich ihr Wert verändert. \end{definition} \begin{definition} @@ -481,13 +555,18 @@ Wegen 13.1 Ana I können Reihen der obigen Form beliebig umgeordnet werden, ohne \index{Maßraum} \index{Maß!endliches} \index{Wahrscheinlichkeitsmaß}\index{Maß!Wahrscheinlichkeits-} -Sei $\fa$ eine $\sigma$-Algebra auf $X$ und $\mu:\fa\to[0,+\infty]$ eine Abbildung. $\mu$ heißt ein \textbf{Maß} auf $\fa$, genau dann wenn gilt: +Sei $\fa$ eine $\sigma$-Algebra auf $X$ und $\mu:\fa\to[0,+\infty]$ +eine Abbildung. $\mu$ heißt ein \textbf{Maß} auf $\fa$, genau dann +wenn gilt: \begin{enumerate} \item[$(M_1)$] $\mu(\emptyset)=0$ -\item[$(M_2)$] Ist $(A_j)$ eine disjunkte Folge in $\fa$, so ist $\mu(\bigcup A_j)=\sum\mu(A_j)$. Diese Eigenschaft heißt \textbf{$\sigma$-Additivität}. +\item[$(M_2)$] Ist $(A_j)$ eine disjunkte Folge in $\fa$, so ist +$\mu(\bigcup A_j)=\sum\mu(A_j)$. Diese Eigenschaft heißt +\textbf{$\sigma$-Additivität}. \end{enumerate} -Ist $\mu$ ein Maß auf $\fa$, so heißt $(X,\fa,\mu)$ ein \textbf{Maßraum}.\\ -Ein Maß $\mu$ heißt \textbf{endlich}, genau dann wenn $\mu(X)<\infty$. Ein Maß $\mu$ heißt ein \textbf{Wahrscheinlichkeitsmaß}, genau dann wenn $\mu(X)=1$ ist. +In diesem Fall heißt $(X,\fa,\mu)$ ein \textbf{Maßraum}.\\ +Ein Maß $\mu$ heißt \textbf{endlich} $:\Leftrightarrow \mu(X)<\infty$.\\ +Ein Maß $\mu$ heißt ein \textbf{Wahrscheinlichkeitsmaß} $:\Leftrightarrow\mu(X)=1$ ist. \end{definition} \begin{beispiel} @@ -495,37 +574,51 @@ Ein Maß $\mu$ heißt \textbf{endlich}, genau dann wenn $\mu(X)<\infty$. Ein Ma \index{Dirac-Maß}\index{Maß!Dirac-} \index{Zählmaß}\index{Maß!Zähl-} \begin{enumerate} -\item Sei $\fa=\cp(X)$ und $x_0\in X$. $\delta_{x_0}:\fa\to[0,+\infty]$ sei definiert durch: -\[\delta_{x_0}(A):= -\begin{cases} -1,\ x_0\in A\\ -0,\ x_0\not\in A -\end{cases}\] -Klar ist, dass $\delta_{x_0}(\emptyset)=0$ ist.\\ -Sei $(A_j)$ eine disjunkte Folge in $\fa$. -\[\delta_{x_0}(\bigcup A_j)= -\left.\begin{cases} -1,\ x_0\in\bigcup A_j\\ -0,\ x_0\not\in\bigcup A_j -\end{cases}\right\}=\sum\delta_{x_0}(A_j)\] -$\delta_{x_0}$ ist ein Maß auf $\cp(X)$ und heißt \textbf{Punktmaß} oder \textbf{Dirac-Maß}. -\item Sei $X:=\mdn$, $\fa:=\cp(X)$ und $(p_j)$ eine Folge in $[0,+\infty]$. Definiere $\mu:\fa\to[0,+\infty]$ durch: -\begin{align*} -\mu(A):= -\begin{cases} -0&,A=\emptyset\\ -\sum_{j\in A}p_j&,A\ne\emptyset -\end{cases} -\end{align*} -Übung: $\mu$ ist ein Maß auf $\fa=\cp(\mdn)$ und heißt ein \textbf{Zählmaß}. Sind alle $p_j=1$, so ist $\mu(A)$ gerade die Anzahl der Elemente von $A$. -\item Sei $(X,\fa,\mu)$ ein Maßraum, $\emptyset\ne Y\subseteq X$ und $\fa_0\subseteq\fa$ eine $\sigma$-Algebra auf $Y$. Definiere $\mu_0:\fa_0\to[0,+\infty]$ durch $\mu_0(A):=\mu(A)$ ($A\in\fa_0$). Dann ist $(Y,\fa_0,\mu_0)$ ein Maßraum.\\ -Ist spezieller $Y\in\fa$, so ist $\fa_0:=\fa_Y\subseteq\fa$ und man definiert $\mu_{|Y}:\fa_Y\to[0,+\infty]$ durch $\mu_{|Y}(A):=\mu(A)$. + \item Sei $\fa:=\cp(X)$ und $x_0\in X$. + $\delta_{x_0}:\fa\to[0,+\infty]$ sei definiert durch: + \[\delta_{x_0}(A):= + \begin{cases} + 1,\ x_0\in A\\ + 0,\ x_0\not\in A + \end{cases}\] + Klar ist, dass $\delta_{x_0}(\emptyset)=0$ ist.\\ + Sei $(A_j)$ eine disjunkte Folge in $\fa$. + \[\delta_{x_0}(\bigcup A_j)= + \left.\begin{cases} + 1,\ x_0\in\bigcup A_j\\ + 0,\ x_0\not\in\bigcup A_j + \end{cases}\right\}=\sum\delta_{x_0}(A_j)\] + $\delta_{x_0}$ ist ein Maß auf $\fa$ und heißt + \textbf{Punktmaß} oder \textbf{Dirac-Maß}. + \item Sei $X:=\mdn$, $\fa:=\cp(X)$ und $(p_j)$ eine Folge in + $[0,+\infty]$. Definiere $\mu:\fa\to[0,+\infty]$ durch: + \begin{align*} + \text{Für } A \in \fa: \quad + \mu(A):= + \begin{cases} + 0 &\text{, falls } A=\emptyset\\ + \sum_{j\in A}p_j &\text{, falls } A\ne\emptyset + \end{cases} + \end{align*} + Übung: $\mu$ ist ein Maß auf $\fa=\cp(\mdn)$ und heißt ein \textbf{Zählmaß}. + Sind alle $p_j=1$, so ist $\mu(A)$ gerade die Anzahl der + Elemente von $A$. + \item Sei $(X,\fa,\mu)$ ein Maßraum, $\emptyset\ne Y\subseteq X$ + und $\fa_0\subseteq\fa$ eine $\sigma$-Algebra auf $Y$. + Definiere $\mu_0:\fa_0\to[0,+\infty]$ durch + $\mu_0(A):=\mu(A)$ ($A\in\fa_0$).\\ + Dann ist + $(Y,\fa_0,\mu_0)$ ein Maßraum.\\ + Ist spezieller $Y\in\fa$, so ist $\fa_0:=\fa_Y\subseteq\fa$ + und man definiert $\mu_{|Y}:\fa_Y\to[0,+\infty]$ durch + $\mu_{|Y}(A):=\mu(A)$ ist ein Maß auf $\fa_Y$. \end{enumerate} \end{beispiel} \begin{satz} \label{Satz 1.7} -\((X,\fa,\mu)\) sei ein Maßraum, es seien \(A,B\in\fa\) und \((A_{j})\) sei eine Folge in \(\fa\). Dann: +\((X,\fa,\mu)\) sei ein Maßraum, es seien \(A,B\in\fa\) und +\((A_{j})\) sei eine Folge in \(\fa\). Dann: \begin{enumerate} \item \(A\subseteq B\,\implies\,\mu(A)\leq\mu(B)\) \item Ist \(\mu(A)<\infty\) und \(A\subseteq B,\implies\,\mu(B\setminus A)=\mu(B)-\mu(A)\) diff --git a/documents/Analysis III/mathe.sty b/documents/Analysis III/mathe.sty index b960d9e..019cc8c 100644 --- a/documents/Analysis III/mathe.sty +++ b/documents/Analysis III/mathe.sty @@ -23,6 +23,8 @@ \usepackage{braket} % needed for \Set \usepackage{ulem} %needed for uwave \usepackage{enumerate} +\usepackage{tikz} +\usetikzlibrary{patterns} %Seitenlayout \setlength{\parindent}{0pt} @@ -75,7 +77,7 @@ \def\cf{\mathcal{F}} \def\ci{\mathcal{I}} \def\co{\mathcal{O}} -\def\cp{\mathcal{P}} +\def\cp{\mathcal{P}} % Potenzmenge \def\T{\mathscr{T}} \def\H{\mathscr{H}} \newcommand{\F}{\mathcal{F}}