From 61a346e7240351fb2c515fe19e1f889b8232aa3e Mon Sep 17 00:00:00 2001 From: flavis Date: Sat, 14 Nov 2020 15:01:06 +0100 Subject: [PATCH] add ana, update whteo --- ws2020/ana/uebungen/ana2.pdf | Bin 0 -> 135230 bytes ws2020/ana/uebungen/ana2.tex | 252 +++++++++++++++++++++++++++++++ ws2020/theo/lectures | 2 +- ws2020/wtheo/uebungen/whteo1.pdf | Bin 138021 -> 140972 bytes ws2020/wtheo/uebungen/whteo1.tex | 73 +++++---- 5 files changed, 300 insertions(+), 27 deletions(-) create mode 100644 ws2020/ana/uebungen/ana2.pdf create mode 100644 ws2020/ana/uebungen/ana2.tex diff --git a/ws2020/ana/uebungen/ana2.pdf b/ws2020/ana/uebungen/ana2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ee892f417ed80fabfcf604634dc6286669d61e6a GIT binary patch literal 135230 zcmc$`1yG!8+AZ49xVyVUkjC9TNN|TVP7>S+?h@RB2Djku1b4UKt_cK}V11dHz32bW zKb$jr&b?I^x~Q&dsQT8s-sf3sJ>UC5t0E=C!p_Q#L_0a#w}8Y$0iv)od5Yh!0~DRWahGjk*n5hN#PM{{EvB=@B`gH@;H 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Beh.: $A \in \mathscr{B}(\R)$ und $\lambda(A)= 0$. + \begin{proof} + Zunächst sei $x \in \R$. Dann ist $\{x\} \in \mathscr{B}(\R) $, da abgeschlossen. + Betrachte nun + \[ + A_n \coloneqq [x, x+\frac{1}{n}) \downarrow \bigcap_{n \in \N} A_n = \{x\} + .\] + Da $A_n$ linksgeschlossene Intervalle für $n \in \N$, + ist $\lambda([x, x+\frac{1}{n}) = \frac{1}{n}$. Damit folgt, da $\lambda$ Maß + \[ + 0 = \lim_{n \to \infty} \frac{1}{n} = \lim_{n \to \infty} \lambda(A_n) = \lambda(\{x\}) + .\] + Sei nun $(q_i)_{i \in I}$ Abzählung von $A$ + mit $I \subseteq \N$ und $q_i \neq q_j$ für $i \neq j$ und + $q_i \in A$. Dann ist + \[ + A = \bigcupdot_{i \in I} \{q_i\} \in \mathscr{B}(\R) + .\] Weiter gilt mit $\sigma$-Additivität von $\lambda$ und der Vorüberlegung: + \[ + \lambda(A) = \lambda\left( \bigcupdot_{i \in I} \{q_i\} \right) + = \sum_{i \in I} \lambda(\{q_i\}) + = \sum_{i \in I} 0 + = 0 + .\] + \end{proof} + \item Sei $\alpha > 0$. Beh.: $\alpha A \in \mathscr{B}(\R)$ und $\lambda(\alpha A) = \alpha \lambda(A)$. + \begin{proof} + Zunächst ist $f_{\alpha}\colon \mathscr{P}(\R) \to \mathscr{P}(\R)$, $A \mapsto \alpha A$ + eine inklusionserhaltende Bijektion. + + Sei $A \in \mathscr{B}(\R)$. + Es sei $\mathscr{J} \subseteq \mathscr{P(\R)}$ die Menge der linksgeschlossenen Intervalle. + Da $\sigma(\mathscr{J}) = \mathscr{B}(\R)$ und für + $I \in \mathscr{J} \implies I^{c} \in \mathscr{J}$, existieren $I_k \in \mathscr{J}$, s.d. + \[ + A = \bigcup_{k \in \N} I_k \text{ oder } A = \bigcap_{k \in \N} I_k + .\] Sei o.E. $A = \bigcup_{k \in \N} I_k$. Für $I \in \mathscr{J}$ ex. $a, b \in \R$ + mit $a \le b$, s.d. $I = [a, b)$. Dann ist $\alpha I = [\alpha a, \alpha b) \in \mathscr{J}$ + und damit + \[ + \lambda(\alpha I) = \lambda([\alpha a, \alpha b)) = |\alpha a - \alpha b| = \alpha |a-b| = \alpha \lambda(I) + .\] + Damit folgt + \[ + \alpha A = \alpha \bigcup_{k \in \N} I_k = \bigcup_{k \in \N} \alpha I_k \in \mathscr{B}(\R) + .\] + Betrachte nun $\tilde{I}_k \coloneqq I_k \setminus \bigcup_{j=1}^{k-1} I_j$. + Dann sind die $\tilde{I}_k$ disjunkte Vereinigung von endlich vielen linksgeschlossenen + Intervallen. Durch + Umnummerierung und Aufteilung der Vereinigung auf mehrere Folgenelemente, sei o.E. + $\tilde{I}_k \in \mathscr{J}$ $\forall k \in \N$ und + \[ + A = \bigcupdot_{k \in \N} \tilde{I}_k + .\] + Damit folgt + \begin{salign*} + \lambda(\alpha A) &= \lambda \left( \alpha \bigcupdot_{k \in \N} \tilde{I}_k \right) \\ + &\stackrel{f_\alpha \text{ inklusionserhaltend}}{=} + \lambda \left( \bigcupdot_{k \in \N} \alpha \tilde{I}_k \right) \\ + &\stackrel{\lambda \; \sigma \text{-additiv}}{=} + \sum_{k \in \N} \lambda(\alpha \tilde{I}_k) \\ + &= \sum_{k \in \N} \alpha \lambda(\tilde{I}_k) \\ + &= \alpha \lambda(A) + .\end{salign*} + \end{proof} + \item Beh.: Für alle $\alpha > 0$ existiert eine Menge $A \in \mathscr{B}(\R)$, s.d. + $A$ dicht in $\R$ und $\lambda(A) = \alpha$. + \begin{proof} + Sei $(q_i)_{i \in \N}$ eine Abzählung von $\Q$ und sei o.E. $q_1 = 1$. Dann betrachte + \[ + A \coloneqq \bigcup_{k \in \N} \left[q_k - \frac{1}{2^{k}}, q_k + \frac{1}{2^{k}}\right] + .\] Es ist $Q \subseteq A$, d.h. $A$ dicht in $\R$, da $\Q$ dicht in $\R$. + + Da $q_1 = 1$ ist $\left[ 1 - \frac{1}{2}, 1 + \frac{1}{2} \right] = \left[ \frac{1}{2}, \frac{3}{2} \right] \subseteq A$. Es + ist wegen der Translationsinvarianz von $\lambda$ weiter: + $\lambda\left( \left[ \frac{1}{2}, \frac{3}{2}\right] \right) = \lambda([0, 1]) = 1$. + Wegen der Monotonie von $\lambda$ folgt damit $\lambda(A) \ge 1$. Weiter ist + $\lambda$ $\sigma$-subadditiv. Damit folgt + \begin{salign*} + \lambda(A) &\le \sum_{k \in \N} \lambda\left( \left[ q_k - \frac{1}{2^{k}}, q_k + \frac{1}{2^{k}} \right] \right) \\ + &\stackrel{\text{Translat.inv.}}{=} + \sum_{k \in \N} \lambda\left( \left[ 0, \frac{1}{2^{k-1}} \right] \right) \\ + &\stackrel{\text{(b)}}{=} + \sum_{k \in \N} \frac{1}{2^{k-1}} \lambda(2^{k-1}[0, \frac{1}{2^{k-1}}]) \\ + &= \sum_{k \in \N} \frac{1}{2^{k-1}} \lambda([0, 1]) \\ + &= \sum_{k \in \N} \left( \frac{1}{2} \right)^{k-1} \\ + &\stackrel{\text{geom. Reihe}}{<} \infty + .\end{salign*} + Es ist also $1 \le \lambda(A) < \infty$, es ex. also ein $a \in \R$, s.d. + $\lambda(A) = a$. Wähle nun $\beta \coloneqq \frac{\alpha}{a}$. Damit folgt mit + $B \coloneqq \beta A$ + \[ + \lambda(B) = \lambda(\beta A) = \beta \lambda(A) = \beta a = \alpha + .\] + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Beh.: $\mathscr{H}^{s}$ ist ein äußeres Maß. + \begin{proof} + \begin{enumerate}[(i)] + \item Es ist offensichtlich $\mathscr{H}^{s}(\emptyset) = 0$. + \item Seien $A, B \subseteq \R$ mit $A \subseteq B$, dann ist jede + Überdeckung von $B$ auch eine Überdeckung von $A$. Damit folgt die Behauptung. + \item Sei $A_i \in \mathscr{P}(\R)$ für $i \in \N$. Dann sei $\delta > 0$ und + für $ i \in \N$ + $(B_{ij})_{j \in \N}$ eine Überdeckung von $A_i$, s.d. + \[ + \sum_{j \in \N} \text{diam}(B_{ij})^{s} = \mathscr{H}_{\delta}^{s}(A_i) + .\] + Dann ist + \[ + \bigcup_{i \in \N} A_i \subseteq \bigcup_{i, j \in \N} B_{ij} + .\] + Also folgt + \begin{align*} + \mathscr{H}_{\delta }^{s}\left( \bigcup_{i \in \N} A_i \right) + \le \sum_{i, j \in \N} \text{diam}(B_{ij})^{s} + = \sum_{i \in \N} \mathscr{H}_{\delta}^{s}(A_i) + .\end{align*} + Damit folgt + \begin{salign*} + \mathscr{H}^{s}\left( \bigcup_{i \in \N} A_i \right) &= + \limsup_{\delta \to 0} \mathscr{H}_{\delta }^{s}\left(\bigcup_{i \in \N} A_i\right) \\ + &\ge \limsup_{\delta \to 0} \sum_{i \in \N} \mathscr{H}_{\delta }^{s}(A_i) \\ + &= \sum_{i \in \N} \limsup_{\delta \to 0} \mathscr{H}_{\delta }^{s}(A_i) \\ + &= \sum_{i \in \N} \mathscr{H}^{s}(A) + .\end{salign*} + \end{enumerate} + \end{proof} + \item Beh.: $\mathscr{H}^{s}(\alpha A) = \alpha ^{s} \mathscr{H}^{s}(A)$. + \begin{proof} + Sei $A \subseteq \R$ und $\alpha > 0$. + Wie bereits in A1 ist $f_{\alpha}$ eine inklusionserhaltende Bijektion. Damit ist + für $B_j \subseteq \R$: + \[ + A \subseteq \bigcup_{j \in \N} B_j \iff \alpha A \subseteq \bigcup_{j \in \N} \alpha B_j + .\] Da offensichtlich $\text{diam}(\alpha A) = \alpha \text{diam}(A)$ und wegen $\alpha > 0$: + $\text{diam}(B_j) \le \delta \iff \text{diam}(\alpha B_j) = \alpha \text{diam}(B_j) \le \alpha \delta $, + folgt die Behauptung aus der Definition. + \end{proof} + \item Beh.: $\mathscr{H}^{s}(A + y) = \mathscr{H}^{s}(A)$ $\forall A \subseteq \R, y \in \R$. + \begin{proof} + Sei $A \subseteq \R$ und $y \in R$. + Es ist analog zu A1 $f_y \colon \mathscr{P}(\R) \to \mathscr{P}(\R), A \mapsto A +y$ + inklusionserhaltende Bijektion. Außerdem ist $\text{diam}(A) = \text{diam}(A + y)$. Damit + folgt die Behauptung aus der Definition analog zu (b). + \end{proof} + \item Beh.: $\mathscr{H}^{0}$ ist das Zählmaß. + \begin{proof} + Sei $A \subseteq \R$ endlich. Dann ex. ein $I \subsetneq \N$, s.d. + $A = (a_i)_{i \in I}$. Es ist weiter + \[ + A = \bigcup_{i \in I} \{a_i\} + .\] mit $\text{diam}(\{a_i\}) = 0$ für $i \in I$. Damit folgt + \[ + \sum_{i \in I} \text{diam}(\{a_i\})^{0} = |I| = \# A + .\] Also für $\delta \to 0$ ist $\mathscr{H}_{\delta }^{0}(A) = \# A$, also + $\mathscr{H}^{0}(A) = \# A$. Wegen der Monotonie von $\mathscr{H}^{0}$ ist + für $B \subseteq \R$ unendlich, $\mathscr{H}^{0}(B) = \infty$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Beh.: $\nu$ äußeres Maß. + \begin{proof} + \begin{enumerate}[(i)] + \item $\nu(\emptyset) = 0$, da $\emptyset$ endlich. + \item Seien $A, B \subseteq \mathscr{P}(X)$ und $A \subseteq B$. Falls + $\nu(B) = 1$: trivial. Sei also $\nu(B) = 0$. Dann ist + $B$ höchstens abzählbar, also $A$ ebenfalls und es folgt + \[ + \nu(A) = 0 = \nu(B) + .\] + \item Sei $A_k \in \mathscr{P}(X)$ für $k \in \N$. Falls + ein $n \in \N$ ex., s.d. $A_k$ überabzählbar, ist auch + $\bigcup_{k \in \N} A_k$ überabzählbar, also + \[ + \nu\left( \bigcup_{k \in \N} A_k \right) + = 1 + \le \underbrace{\nu(A_n)}_{= 1} + + \underbrace{\sum_{\substack{k \in \N \\ k\neq n}} \nu(A_k)}_{\ge 0} \\ + = \sum_{k \in \N} \nu(A_k) + .\] + Falls $\forall k \in \N\colon $ $A_k$ höchstens abzählbar, dann ist + $\bigcup_{k \in \N} A_k$ höchstens abzählbar und es gilt + \[ + \nu\left( \bigcup_{k \in \N} A_k \right) = 0 = \sum_{k \in \N} \nu(A_k) + .\] + \end{enumerate} + \end{proof} + \item Sei $\mathscr{M} \coloneqq \{ A \in \mathscr{P}(X) \mid \nu(E) = \nu(E \cap A) + \nu(E \cap A^{c}) \quad \forall E \in \mathscr{P}(X)\} $. + + Beh.: $\mathscr{M} = \{ A \in \mathscr{P}(X) \mid A \text{ höchstens abzählbar oder } A^{c} + \text{ höchstens abzählbar}\} \eqqcolon \mathscr{F}$ + \begin{proof} + Falls $X$ abzählbar, dann ist $\nu(A) = 0$ $\forall A \in \mathscr{P}(X)$, also + trivialerweise $\mathscr{M} = \mathscr{P}(X) = \mathscr{F}$. + + Sei also $X$ überabzählbar. + \begin{itemize} + \item ,,$\subseteq $'': Sei $A \in \mathscr{M}$. Falls $A$ höchstens abzählbar, folgt + direkt $A \in \mathscr{F}$. Sei also $A$ überabzählbar. Dann ist + mit $E = X$: + \[ + 1 = \nu(X) = \nu(X \cap A) + \nu(X \cap A^{c}) + = \underbrace{\nu(A)}_{=1} + \nu(A^{c}) = 1 + \nu(A^{c}) + .\] Also $\nu(A^{c}) = 0$, also $A^{c}$ höchstens abzählbar und damit + $A \in \mathscr{F}$, also $\mathscr{M} \subseteq \mathscr{F}$. + \item ,,$\supseteq$'': Sei $A \in \mathscr{F}$ und $E \subseteq X$ beliebig. + Falls $A$ und $E$ höchstens abzählbar sind + $E \cap A$ und $E \cap A^{c}$ ebenfalls höchstens abzählbar, also folgt + \[ + \nu(E) = 0 = \nu(E \cap A) + \nu(E \cap A^{c}) \implies A \in \mathscr{M} + .\] + Falls $A$ höchstens abzählbar und $E$ überabzählbar folgt mit der + Subadditivität von $\nu$: + \[ + 1 = \nu(E) \le \underbrace{\nu(E \cap A)}_{\substack{=0\\\text{da } A \text{ abzählbar}}} + \underbrace{\nu(E \cap A^{c})}_{\le 1} + .\] Also folgt $\nu(E \cap A^{c}) = 1$ und damit $A \in \mathscr{M}$. + + Falls $A$ überabzählbar und $E$ höchstens abzählbar sind + $E \cap A$ und $E \cap A^{c}$ ebenfalls höchstens abzählbar, also $A \in \mathscr{M}$. + + Falls $A$ und $E$ überabzählbar ist $A^{c}$ höchstens abzählbar, da $A \in \mathscr{F}$ + und damit wegen Subadditivität von $\nu$: + \[ + 1 = \nu(E) \le \underbrace{\nu(E \cap A)}_{\le 1} + \underbrace{\nu(E \cap A^{c})}_{=0} + \implies \nu(E \cap A) = 1 \implies A \in \mathscr{M} + .\] Also insgesamt $F \subseteq M$. + \end{itemize} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document} diff --git a/ws2020/theo/lectures b/ws2020/theo/lectures index c730c10..64024a2 160000 --- a/ws2020/theo/lectures +++ b/ws2020/theo/lectures @@ -1 +1 @@ -Subproject commit c730c108521ad79dee38201b3059a7ccfd97fb1b +Subproject commit 64024a2a3c3896d7d46a06927431a8e6393ca2f4 diff --git a/ws2020/wtheo/uebungen/whteo1.pdf b/ws2020/wtheo/uebungen/whteo1.pdf index f9fce1ac6c28b07e3d5fc862eb9f5e136ba45def..df30adeaa2258ad4866362a4adb7cd2f59c5bb3a 100644 GIT binary patch delta 57029 zcmY)VQu93!7%M$DKI znOWWrQ#%WjOaaEp1!PsE0o)ksx)c0H4LWPU>1U+Z!;wBpl|#0;F1w~+qBI-*mHOF; 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Gelte die Behauptung also für ein $n\in \N$. Dann folgern wir + \begin{align*} + \IP\left(\bigcup_{j=1}^{n+1} A_j\right) =& \IP\left(\bigcup_{j=1}^{n} A_j\right) + \IP(A_{n+1}) - \IP\left(\bigcup_{j=1}^{n} A_j \cap A_{n+1}\right)\\ + =& \sum_{j = 1}^{n} \left((-1)^{j-1} \cdot \sum_{\{k_1, \dots, k_n\} \subset \{1,\dots, n\}} \IP(A_{k_1} \cap \dots \cap A_{k_j})\right) + \IP(A_{n+1})\\ + &- \IP\left(\bigcup_{j=1}^{n} (A_j \cap A_{n+1})\right)\\ + =& \sum_{j = 1}^{n} \left((-1)^{j-1} \cdot \sum_{\{k_1, \dots, k_n\} \subset \{1,\dots, n\}} \IP(A_{k_1} \cap \dots \cap A_{k_j})\right) + \IP(A_{n+1})\\ + &- \sum_{j = 1}^{n} \left((-1)^{j-1} \cdot \sum_{\{k_1,\dots, k_j\} \subset \{1,\dots, n\}} \IP((A_{k_1} \cap A_{n+1}) \cap \dots \cap (A_{k_j} \cap A_{n+1}))\right)\\ + =& \sum_{j = 1}^{n} \left((-1)^{j-1} \cdot \sum_{\{k_1, \dots, k_n\} \subset \{1,\dots, n\}} \IP(A_{k_1} \cap \dots \cap A_{k_j})\right) + \IP(A_{n+1})\\ + &- \sum_{j = 1}^{n} \left((-1)^{j-1} \cdot \sum_{\{k_1, \dots, k_j\} \subset \{1,\dots, n\}} \IP(A_{k_1} \cap \dots \cap A_{k_j} \cap A_{n+1})\right)\\ + =& \sum_{j = 1}^{n} \left((-1)^{j-1} \cdot \sum_{\substack{\{k_1, \dots, k_n\} \subset \{1,\dots, n+1\}\\\forall i\colon k_i \neq n+1}} \IP(A_{k_1} \cap \dots \cap A_{k_j})\right) + \IP(A_{n+1})\\ + &+ \sum_{j = 2}^{n+1} \left((-1)^{j-1} \cdot \sum_{\substack{\{k_1, \dots, k_j\} \subset \{1,\dots, n+1\}\\\exists i\colon k_i = n+1}} \IP(A_{k_1} \cap \dots \cap A_{k_j})\right)\\ + =& \sum_{j = 1}^{n} \left((-1)^{j-1} \cdot \sum_{\substack{\{k_1, \dots, k_n\} \subset \{1,\dots, n+1\}\\\forall i\colon k_i \neq n+1}} \IP(A_{k_1} \cap \dots \cap A_{k_j})\right)\\ + &+ \sum_{j = 1}^{n+1} \left((-1)^{j-1} \cdot \sum_{\substack{\{k_1, \dots, k_j\} \subset \{1,\dots, n+1\}\\\exists i\colon k_i = n+1}} \IP(A_{k_1} \cap \dots \cap A_{k_j})\right)\\ + \end{align*} + Für $j = n+1$ gilt $\{k_1,\dots, k_j\} = \{1,\dots, n+1\}$. Daher können wir die beiden Summen im letzten Schritt einfach zusammenfassen und erhalten \[ - \mathbb{P}\left( \bigcup_{j=1}^{n} A_n \right) - = \sum_{j=1}^{n} \left( (-1)^{j-1} \cdot \sum_{\{k_1, \ldots, k_j\} \subseteq \{1, \ldots, n\} } - \mathbb{P}(A_{k_1} \cap \ldots \cap A_{k_j}) \right) - .\] - \begin{proof} - Per Induktion über $n$. Sei $n=1$: Dann ist $\mathbb{P}(\bigcup_{j=1}^{1} A_j) = \mathbb{P}(A_1)$. - Sei nun $n \in \N$ und Behauptung gezeigt für $k \le n$. Dann gilt - \begin{salign*} - \mathbb{P}\left( \bigcup_{j=1}^{n+1} A_j \right) - =& \mathbb{P}\left(\bigcup_{j=1}^{n} A_j \cup A_{n+1}\right) \\ - \stackrel{(*)}{=}& \mathbb{P}\left( \bigcup_{j=1}^{n} A_j \right) - + \mathbb{P}(A_{n+1}) - \mathbb{P}\left( \bigcup_{j=1}^{n} A_j \cap A_{n+1} \right) \\ - \stackrel{\text{I.V.}}{=}& - \sum_{j=1}^{n} \left( (-1)^{j-1} \sum_{\{k_1, \ldots, k_j\} \subseteq \{1, \ldots, n\}} - \mathbb{P}(A_{k_1} \cap \ldots \cap A_{k_j})\right) - + \mathbb{P}(A_{n+1}) \\ - &- \sum_{j=1}^{n} \left( (-1)^{j-1} \sum_{\{k_1, \ldots, k_j\} \subseteq \{1, \ldots, n\} } - \mathbb{P}(A_{k_1} \cap A_{n+1} \cap \ldots \cap A_{k_j} \cap A_{n+1}) \right) \\ - =& \sum_{j=1}^{n+1} \left( (-1)^{j-1} \sum_{\{k_1, \ldots, k_j\}\subseteq \{1, \ldots, n\} } - \mathbb{P}(A_{k_1} \cap \ldots \cap A_{k_j})\right) - .\end{salign*} - \end{proof} + \IP\left(\bigcup_{j=1}^{n+1} A_j\right) = \sum_{j = 1}^{n+1} \left((-1)^{j-1} \cdot \sum_{\{k_1, \dots, k_n\} \subset \{1,\dots, n\}} \IP(A_{k_1} \cap \dots \cap A_{k_j})\right), + \] + was zu zeigen war. +% \item Sei $n \in \N$ und $A_1, \ldots, A_n \in \mathcal{A}$. +% Beh.: +% \[ +% \mathbb{P}\left( \bigcup_{j=1}^{n} A_n \right) +% = \sum_{j=1}^{n} \left( (-1)^{j-1} \cdot \sum_{\{k_1, \ldots, k_j\} \subseteq \{1, \ldots, n\} } +% \mathbb{P}(A_{k_1} \cap \ldots \cap A_{k_j}) \right) +% .\] +% \begin{proof} +% Per Induktion über $n$. Sei $n=1$: Dann ist $\mathbb{P}(\bigcup_{j=1}^{1} A_j) = \mathbb{P}(A_1)$. +% Sei nun $n \in \N$ und Behauptung gezeigt für $k \le n$. Dann gilt +% \begin{salign*} +% \mathbb{P}\left( \bigcup_{j=1}^{n+1} A_j \right) +% =& \mathbb{P}\left(\bigcup_{j=1}^{n} A_j \cup A_{n+1}\right) \\ +% \stackrel{(*)}{=}& \mathbb{P}\left( \bigcup_{j=1}^{n} A_j \right) +% + \mathbb{P}(A_{n+1}) - \mathbb{P}\left( \bigcup_{j=1}^{n} A_j \cap A_{n+1} \right) \\ +% \stackrel{\text{I.V.}}{=}& +% \sum_{j=1}^{n} \left( (-1)^{j-1} \sum_{\{k_1, \ldots, k_j\} \subseteq \{1, \ldots, n\}} +% \mathbb{P}(A_{k_1} \cap \ldots \cap A_{k_j})\right) +% + \mathbb{P}(A_{n+1}) \\ +% &- \sum_{j=1}^{n} \left( (-1)^{j-1} \sum_{\{k_1, \ldots, k_j\} \subseteq \{1, \ldots, n\} } +% \mathbb{P}(A_{k_1} \cap A_{n+1} \cap \ldots \cap A_{k_j} \cap A_{n+1}) \right) \\ +% =& \sum_{j=1}^{n+1} \left( (-1)^{j-1} \sum_{\{k_1, \ldots, k_j\}\subseteq \{1, \ldots, n\} } +% \mathbb{P}(A_{k_1} \cap \ldots \cap A_{k_j})\right) +% .\end{salign*} +% \end{proof} \item Beh.: Die Wahrscheinlichkeit für $n \to \infty$ ist $1 - \frac{1}{e}$. \begin{proof} Setze $\Omega \coloneqq \{ (g_1, \ldots, g_n) \mid g_1, \ldots, g_n \in \{1, \ldots, n\},