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\underbrace{\nu(E \cap A)}_{\le 1} + \underbrace{\nu(E \cap A^{c})}_{=0} \implies \nu(E \cap A) = 1 \implies A \in \mathscr{M} - .\] Also insgesamt $F \subseteq M$. + .\] Also insgesamt $\mathscr{F} \subseteq \mathscr{M}$. \end{itemize} \end{proof} \end{enumerate} \end{aufgabe} +\begin{aufgabe} + Beh.: $\mu$ ist ein Maß. + \begin{proof} + \begin{enumerate}[(i)] + \item $\mu(\emptyset) = 0$, denn $\#(\emptyset \cap \{1, \ldots, n\}) = 0$ $\forall n \in \N$. + \item Sei $A_i \in \mathscr{P}(\N)$ für $i \in \N$ und $A_i \cap A_j = \emptyset$ für $i\neq j$. + Dann gilt + \begin{salign*} + \mu\left( \bigcupdot_{i \in \N} A_i \right) + &= \limsup_{n \to \infty} \frac{1}{n} \# \left( \bigcupdot_{i \in \N} A_i \cap \{1, \ldots, n\} \right) \\ + &= \limsup_{n \to \infty} \frac{1}{n} \# \left( \bigcupdot_{i \in \N} (A_i \cap \{1, \ldots, n\} ) \right) \\ + &\stackrel{\text{disj. Ver.}}{=} \limsup_{n \to \infty} \frac{1}{n} \sum_{i \in \N} \#(A_i \cap \{1, \ldots, n\}) \\ + &= \sum_{i \in \N} \limsup_{n \to \infty} \frac{1}{n} \#(A_i \cap \{1, \ldots, n\}) \\ + &= \sum_{i \in \N} \mu(A_i) + .\end{salign*} + \end{enumerate} + \end{proof} +\end{aufgabe} + \end{document}