From d636dfeb3f4a81c46f1508d4f9af2b2537723a00 Mon Sep 17 00:00:00 2001 From: flavis Date: Tue, 24 Nov 2020 10:26:38 +0100 Subject: [PATCH] update ana3 --- ws2020/ana/uebungen/ana3.pdf | Bin 165012 -> 171661 bytes ws2020/ana/uebungen/ana3.tex | 41 +++++++++++++++++++++++++++++++++++ 2 files changed, 41 insertions(+) diff --git a/ws2020/ana/uebungen/ana3.pdf b/ws2020/ana/uebungen/ana3.pdf index 9affc609ef57219e59da91a11fca09d97171a2bb..669c109c0340bdd0ff87e7b0bb98426af593d2a5 100644 GIT binary patch delta 55916 zcmX7tWmFwavxPah>j8oVcekJ+K+xds?(QxJmmmQKcXxLuSb*Tc-Q6uX+`QlY)q8q+ zt*Kqr{Y=kvVkB}*7jhgGG#4LGXG>Qnj|PI_3iAT=L3)szN_t$2H&Z#eN-N@-gPJs! zqqy2Y29Wc|PdF(r>6+}GG!t4^A)$uQX9H$Tg;Y3DIWmTJuqw4-Q7Y;QCnwKm_u(JU z(>j@Y(Ne~o3eQpYOUv#O_+n>0Mu;W7+YMvC=O!A8oY2E2m2vJV%%<(6dug)aRDKIG z)T6IS4ibl8^G*DrVTx1x)o-x9!WzYm9>z|;0K#r{A7XcCZ7d4&=}#&Tv#3Y394gzn zRIBS+uYA7-Q@t`*+?qO#W~clx(6wWHMQl&dPKPNVim&o(rx+&HVr^x#Kj(2sobUMC zt0uLsC4K$@;j*gGWL~v?_hx&@xa%S`)tTw%rF9d8wc0WOuE-_8%1i)c4tfuS)_l~y 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zn-a+{ZK%)BID@y|TPQ5!u9X3!B>rE8^1P^0B-&Kx_7&K&DC`(!^CZYoGeturF=UA; z4iT-caym5g$IRC05k2EYX7M)+i78PCe={3J8rfHh(B{%NwQD>qUa{;pZ8_y%r_2`~5`5SqEM{eM=DCN?mxZpJQdzg)~MV3;}Bm^nFN$jQYO HC1Cy!t|*<9 diff --git a/ws2020/ana/uebungen/ana3.tex b/ws2020/ana/uebungen/ana3.tex index 1587028..83720c0 100644 --- a/ws2020/ana/uebungen/ana3.tex +++ b/ws2020/ana/uebungen/ana3.tex @@ -368,4 +368,45 @@ \end{enumerate} \end{aufgabe} +\newpage + +\begin{aufgabe} + Beh.: $f$ genau dann messbar, wenn $f^{-1}(\mathscr{A}) \subset \mathscr{E}$. + \begin{proof} + ,,$\implies$'': trivial, denn $f$ messbar $\implies$ $f^{-1}(\mathscr{F}) \subset \mathscr{E}$ und da $\mathscr{A} \subset \mathscr{F}$, folgt + $f^{-1}(\mathscr{A}) \subset \mathscr{E}$. + + ,,$\impliedby$'': Sei also $f^{-1}(\mathscr{F}) \subset \mathscr{E}$. + Also $f^{-1}(\mathscr{F}) = \{ f^{-1}(A) \mid A \in \mathscr{F} \} \subset \mathscr{E}$. + \[ + \mathscr{K} := \{ A \in \mathscr{F} \mid f^{-1}(A) \in \mathscr{E}\} + .\] + + Z.z.: $\mathscr{K}$ $\sigma$-Algebra. + \begin{enumerate}[(i)] + \item $Y \in \mathscr{K}$, denn + $f^{-1}(Y) = X \in \mathscr{E}$, da $\mathscr{E}$ + $\sigma$-Algebra. + \item Sei $A \in \mathscr{A}$. Dann ist + $f^{-1}(A) \in \mathscr{E}$ und damit + $f^{-1}(A^{c}) = f^{-1}(A)^{c} \in \mathscr{E}$, da + $\mathscr{E}$ $\sigma$-Algebra. + \item Seien $A_i \in \mathscr{K}$ für $i \in \N$. Dann + ist $\forall i \in \N$: $f^{-1}(A_i) \in \mathscr{E}$. Damit + folgt, da $\mathscr{E}$ $\sigma$-Algebra: + \[ + f^{-1}\left(\bigcup_{i \in \N} A_i \right) + = \bigcup_{i \in \N} f^{-1}(A_i) \in \mathscr{E} + .\] + \end{enumerate} + Nach Voraussetzung ist $\mathscr{A} \subset \mathscr{K}$. Es + ist $\mathscr{K} \subset \mathscr{F}$ und + $\mathscr{K}$ $\sigma$-Algebra, die $\mathscr{A}$ enthält, damit + folgt $\mathscr{F} = \sigma(\mathscr{A}) \subset \mathscr{K}$, + also insgesamt $\mathscr{K} = \mathscr{F}$. Also + folgt $\forall A \in \mathscr{F}\colon f^{-1}(A) \in \mathscr{E}$, also + $f^{-1}(\mathscr{F}) \subset \mathscr{E}$. + \end{proof} +\end{aufgabe} + \end{document}