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-\documentclass{../../../lecture}
-
-\usepackage[]{mathrsfs}
-
-\begin{document}
-
-\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}