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\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}