From 997f2545ec090442754dc3df21b16f75fc9ac560 Mon Sep 17 00:00:00 2001 From: flavis Date: Wed, 18 Nov 2020 20:32:07 +0100 Subject: [PATCH] rework ana, update wtheo --- ws2020/ana/uebungen/ana2.pdf | Bin 136600 -> 141513 bytes ws2020/ana/uebungen/ana2.tex | 117 +++++++++++++++++++++-------------- ws2020/wtheo/uebungen | 2 +- 3 files changed, 70 insertions(+), 49 deletions(-) diff --git a/ws2020/ana/uebungen/ana2.pdf b/ws2020/ana/uebungen/ana2.pdf index 134d18059361f17510dd076d1055dc984edbb33d..50e599ef22895c598121dee666d95616c11c0733 100644 GIT binary patch delta 102191 zcmZshWl-KR)30%NZmf85E$;5_Qrz9OP$)O<{>NR5ySuwP6e;djwD{@syz_oJ=X}a$ zGMUUKzhrXlZdwJx*c?JU1r!@g>dpxy9r%K1J#k_5{ztc%hM@q?;$4@XeFFXHIV`;j z%+ExP&>z<<{S+)2x)TG;nft%pbXOctMw{&i?A2ZZIXx?8PTYDp(Wi~N%;ySt^5;Gm z%stX5Pb-u7j%XysOSlnd_e4?BL`tV1_D4rd*Azsp?fG`U>YESQFb1bZgq74Kxq=_0 z5WKxJbBFDlc`O?G1@)PCY`P6Ql3j;)7#Pa3->ACADpGSds&Q|ZmflWlgpF$IO|z$N 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z6J$+!W=ZnDq+|$o z=y;G302D5`7rn}c4r=YppgqwGoagE9q(o8qipe)v3P&l3+E_`HLM1nNfG34$_1uOnK!Nhvy;MZx+a*1xpwX@Gi|EMpt3?D`Tz^0L7(* zhJHVY93;&mjUl66K&sWc#h<@9aAw+8VCB?-IiV2(-`bY^!I>uB=ZZc`I3&0J<~vjerVjtLej#eMm5sd5!-j-T;yDjV_Sm1>or z=1O&C&BFz3W%3cS(u#PW21;cohZN+6r9ObctX5n;wmEuZp!ygf#o>pXoI?qQhrq(d O!NCPbMkXdN4)v diff --git a/ws2020/ana/uebungen/ana2.tex b/ws2020/ana/uebungen/ana2.tex index e5c58d0..79bb802 100644 --- a/ws2020/ana/uebungen/ana2.tex +++ b/ws2020/ana/uebungen/ana2.tex @@ -37,43 +37,65 @@ \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 + Betrachte \[ - \alpha A = \alpha \bigcup_{k \in \N} I_k = \bigcup_{k \in \N} \alpha I_k \in \mathscr{B}(\R) + \mathscr{D} = \{ A \in \mathscr{B}(\R) \mid \alpha A \in \mathscr{B}(\R) \} .\] - 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 + Dann ist $\mathscr{D}$ Dynkinsystem, denn + \begin{enumerate}[(i)] + \item $\R \in \mathscr{D}$, denn $\alpha \R = \R$. + \item Sei $A \in \mathscr{D}$. Dann ist $\alpha A \in \mathscr{B}(\R)$, also + \[ + \alpha A^{c} = (\alpha A)^{c} \in \mathscr{B}(\R) + .\] Also $A^{c} \in \mathscr{D}$. + \item Sei $A_i \in \mathscr{D}$ $\forall i \in \N$ mit $A_i \cap A_j = \emptyset$. Dann + ist + \[ + \alpha \bigcupdot_{i \in \N} A_i = \bigcup_{i \in \N} \underbrace{\alpha A_i}_{\in \mathscr{B}(\R)} \in \mathscr{B}(\R) + .\] Also $\bigcupdot_{i \in \N} A_i \in \mathscr{D} $. + \end{enumerate} + Sei $\mathscr{J} \subseteq \mathscr{P}(\R)$ die Menge der linksgeschlossenen Intervalle. + Es ist offensichtlich $\mathscr{J} \subseteq \mathscr{D}$ und $\mathscr{J}$ + $\pi$-System. Da auch + $\sigma(\mathscr{J}) = \mathscr{B}(\R)$ folgt mit ÜB 1, dass \[ - A = \bigcupdot_{k \in \N} \tilde{I}_k + \mathscr{B}(\R) = \sigma(\mathscr{J}) \subseteq \mathscr{D} .\] - 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*} + + Es ist $f_{\alpha}\colon \mathscr{P}(\R) \to \mathscr{P}(\R)$, $A \mapsto \alpha A$ + eine inklusionserhaltende Bijektion. Das heißt, die Disjunktheit von Mengen bleibt erhalten + $(*)$. Damit ist + \[ + \mathscr{H} = \{ A \in \mathscr{B}(\R) \mid \lambda(\alpha A) = \alpha \lambda(A)\} + \] ein Dynkinsystem, denn + \begin{enumerate}[(i)] + \item $\R \in \mathscr{H}$, denn $\lambda(\R) = \lambda(\alpha \R) = \alpha \lambda(\R)$. + \item Sei $A \in \mathscr{H}$. Dann ist + \begin{salign*} + \alpha \lambda(A^{c}) &= \alpha \left[ \lambda(\R) - \lambda(A) \right] \\ + &= \lambda(\alpha \R) - \lambda (\alpha A) \\ + &= \lambda((\alpha A)^{c} \\ + &= \lambda(\alpha A^{c}) + .\end{salign*} + Also $A^{c} \in \mathscr{H}$. + \item Seien $A_i \in \mathscr{H}$ $\forall i \in \N$, $A_i \cap A_j = \emptyset$ für + $i \neq j$. Dann ist + \begin{salign*} + \lambda\left(\alpha\bigcupdot_{i \in \N} A_i\right) &\stackrel{(*)}{=} + \lambda\left( \bigcupdot_{i \in \N} \alpha A_i \right) \\ + &\stackrel{\lambda \text{ Maß}}{=} + \sum_{i \in \N} \lambda(\alpha A_i) \\ + &\stackrel{A_i \in \mathscr{H}}{=} + \alpha \sum_{i \in \N} \lambda(A_i) \\ + &= \alpha \bigcupdot_{i \in \N} A_i + .\end{salign*} + \end{enumerate} + Für $I \in \mathscr{J}$ gilt offensichtlich für $a, b \in \R$ mit $b \ge a$: + \[ + \lambda(\alpha I) = \lambda(\alpha [a, b)) = \lambda([\alpha a, \alpha b)) + = |\alpha a - \alpha b| = \alpha |a - b| = \alpha \lambda([a, b)) = \alpha \lambda(I) + .\] Also $\mathscr{J} \subseteq \mathscr{H}$. Dann folgt analog zu oben + $\mathscr{B}(\R) \subseteq \mathscr{H}$. \end{proof} \item Beh.: Für alle $\alpha > 0$ existiert eine Menge $A \in \mathscr{B}(\R)$, s.d. $A$ dicht in $\R$ und $\lambda(A) = \alpha$. @@ -145,7 +167,7 @@ \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 + $f_{\alpha}$ ist 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 @@ -250,21 +272,20 @@ \end{aufgabe} \begin{aufgabe} - Beh.: $\mu$ ist ein Maß. + Beh.: $\mu$ ist weder Maß noch äußeres 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} + Betrachte $A_n \coloneqq \{n\}$ für $n \in \N$. Dann ist + \[ + \bigcup_{n \in \N} A_n = \bigcup_{n \in \N} \{n\} = \N + .\] + Dann ist für $k \in \N$: + \[ + \mu(A_k) = \limsup_{n \to \infty} \frac{1}{n} \# ( \{k\} \cap \{1, \ldots, n\}) = 0 + ,\] aber + \[ + \mu(\N) = \limsup_{n \to \infty} \frac{1}{n} \# (\N \cap \{1, \ldots, n\}) = \limsup_{n \to \infty} \frac{n}{n} = 1 + > \sum_{n \in \N} A_n + .\] Also ist $\mu$ nicht subadditiv, also weder Maß noch äußeres Maß. \end{proof} \end{aufgabe} diff --git a/ws2020/wtheo/uebungen b/ws2020/wtheo/uebungen index 5c0c1a5..9915d40 160000 --- a/ws2020/wtheo/uebungen +++ b/ws2020/wtheo/uebungen @@ -1 +1 @@ -Subproject commit 5c0c1a5904f9777dbadfc9f48d47f17c3b212679 +Subproject commit 9915d40c31f5fa3301c178833217f0d64f2b488d