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100644 index 0000000..b54c840 --- /dev/null +++ b/wtheo3.tex @@ -0,0 +1,160 @@ +\documentclass[uebung]{../../../lecture} + +\title{Wtheo 0: Übungsblatt 3} +\author{Josua Kugler, Christian Merten} +\usepackage[]{bbm} +\usepackage[]{mathrsfs} + +\begin{document} + +\punkte[9] + +\begin{aufgabe} + + \begin{enumerate}[(a)] + \item Beh.: $C_{\alpha, x_m} = \alpha x_m^{\alpha}$. + \begin{proof} + Es muss gelten + \begin{salign*} + \int_{-\infty}^{\infty} \mathbbm{f}(x) \d x &= 1 + \intertext{Damit folgt} + \int_{-\infty}^{\infty} C_{\alpha, x_m} x^{-(\alpha +1)} \mathbbm{1}_{\{x \ge x_m\} } \d x + &= \int_{x_m}^{\infty} C_{\alpha,x_m} x^{-(\alpha + 1)} \d x \\ + &= C_{\alpha, x_m} \int_{x_m}^{\infty} x^{-(\alpha +1)} \d x \\ + &\stackrel{\alpha + 1 > 1}{=} - C_{\alpha, x_m} \frac{1}{\alpha} x^{-\alpha} \Big|_{x_m}^{\infty} \\ + &= - \frac{C_{\alpha,x_m}}{\alpha} \lim_{z \to \infty} \left[ z^{-\alpha} - x_m^{-\alpha} \right] \\ + &\stackrel{\alpha > 0}{=} \frac{C_{\alpha, x_m}}{\alpha} x_m^{-\alpha} \\ + &\stackrel{!}{=} 1 + \intertext{Damit folgt dann} + C_{\alpha,x_m} &= \alpha x_m^{\alpha} + .\end{salign*} + \end{proof} + \item Beh.: $\mathbbm{F}(x) = \left( 1 - \left( \frac{x_m}{x} \right)^{\alpha} \right) \mathbbm{1}_{\{x \ge x_m > 0\} }$ + \begin{proof} + Falls $x < x_m$ ist $\mathbbm{f}(y) = 0$ $\forall y \le x$, also $\mathbb{F}(x) = 0$. + Sei also $x \ge x_m$. Dann folgt + \begin{salign*} + \mathbb{F}(x) &= \int_{x_m}^{x} \alpha x_m^{\alpha} y^{-(\alpha + 1)} \d y \\ + &= - x_m^{\alpha} y^{-\alpha} \Big|_{x_m}^{x} \\ + &= - x_m^{\alpha} \left[ x^{-\alpha} - x_m^{-\alpha} \right] \\ + &= -x_m^{\alpha} x^{-\alpha} + x_m^{\alpha} x_m^{-\alpha} \\ + &= 1 - \left( \frac{x_m}{x} \right)^{\alpha} + \intertext{Insgesamt folgt} + \mathbb{F}(x) &= \left( 1 - \left( \frac{x_m}{x} \right)^{\alpha} \right) \mathbbm{1}_{\{x \ge x_m > 0\} } + .\end{salign*} + \end{proof} + \item Beh.: $\mathbb{P}([1,2]) = \frac{1}{2} = \mathbb{P}((2, \infty))$. + \begin{proof} + Mit $\alpha = x_m = 1$ folgt $\mathbb{F}(x) = \left( 1 - \frac{1}{x} \right) \mathbbm{1}_{\{x \le 1\} }$. Damit folgt + \begin{salign*} + \mathbb{P}([1,2]) &= \mathbb{F}(2) - \mathbb{F}(1) = 1 - \frac{1}{2} - 1 + 1 = \frac{1}{2} \\ + \mathbb{P}((2, \infty)) &= 1 - \mathbb{P}((-\infty, 2]) = 1 - \mathbb{F}(2) = 1 - 1 +\frac{1}{2} = \frac{1}{2} + .\end{salign*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\stepcounter{aufgabe} + +\begin{aufgabe} + Zunächst ist zu bemerken, dass mit $\mathscr{E} := \{ (a, \infty] \mid a \in \R\} $ nach VL + gilt $\sigma(\mathscr{E}) = \overline{\mathscr{B}}$. Damit ist + $f\colon \Omega \to \overline{\R}$ genau dann $(\mathscr{A}, \overline{\mathscr{B}})$ messbar, + wenn $f^{-1}(\mathscr{E}) \subseteq \mathscr{A}$. + \begin{enumerate}[(a)] + \item + \begin{enumerate}[(1)] + \item Sei $m \in \N$. Beh.: Folgende Abbildungen sind $(\mathscr{A}, \overline{\mathscr{B}})$ + messbar: + \begin{enumerate}[(i)] + \item $\sup_{n \ge m} X_n \colon \Omega \to \overline{\R}$ + \item $\inf_{n \ge m} X_n \colon \Omega \to \overline{\R}$ + \end{enumerate} + \begin{proof} + \begin{enumerate}[(i)] + \item Sei $a \in \R$ bel. Für $x \in \R$ gilt dann + \[ + \sup_{n \ge m} X^{n}(x) > a \iff \exists n \ge m\colon X^{n}(x) > a + .\] + Damit folgt da $X^{n}$ $(\mathscr{A}, \overline{\mathscr{B}})$-messbar + und $\mathscr{A}$ $\sigma$-Algebra: + \begin{salign*} + (\sup_{n \ge m} X_n)^{-1}((a, \infty])) + &= \{ x \in \Omega \mid \sup_{n \ge m} X^{n}(x) > a\} \\ + &= \{ x \in \Omega \mid \exists n \ge m\colon X^{n} > a\} \\ + &= \bigcup_{n \ge m} \{ x \in \Omega \mid X^{n}(x) > a\} \\ + &= \bigcup_{n \ge m} \underbrace{(X^{n})^{-1}((a, \infty])}_{\in \mathscr{A}} + \in \mathscr{A} + .\end{salign*} + Also $\sup_{n \ge m} X^{n}$ $(\mathscr{A}, \overline{\mathscr{B}})$-messbar. + \item Sei $a \in \R$. Für $x \in \R$ gilt dann + \[ + \inf_{n \ge m} X^{n}(x) < a \iff \exists n \ge m\colon X^{n}(x) < a + .\] + Damit folgt da $X^{n}$ $(\mathscr{A}, \overline{\mathscr{B}})$-messbar + und $\mathscr{A}$ $\sigma$-Algebra: + \begin{salign*} + (\inf_{n \ge m} X_n)^{-1}([-\infty, a)) + &= \{ x \in \Omega \mid \inf_{n \ge m} X^{n}(x) < a\} \\ + &= \{ x \in \Omega \mid \exists n \ge m\colon X^{n} < a\} \\ + &= \bigcup_{n \ge m} \{ x \in \Omega \mid X^{n}(x) < a\} \\ + &= \bigcup_{n \ge m} \underbrace{(X^{n})^{-1}([-\infty, a))}_{\in \mathscr{A}} + \in \mathscr{A} + .\end{salign*} + Da auch $\sigma(\{ [-\infty, a) \mid a \in \R \}) = \overline{\mathscr{B}}$ + folgt + also $\inf_{n \ge m} X^{n}$ $(\mathscr{A}, \overline{\mathscr{B}})$-messbar. + \end{enumerate} + \end{proof} + \item Beh.: $\limsup_{n \to \infty} X^{n}$ und $\liminf_{n \to \infty} X^{n}$ sind + $(\mathscr{A}, \overline{\mathscr{B}})$-messbar. + + \begin{proof} + Definiere $f_m \coloneqq \sup_{n \ge m} X_n$. Dann ist $f_m$ messbar nach (1)(i) + und + \[ + \limsup_{n \to \infty} X_n = \inf_{m \ge 1} \sup_{n \ge m} X_n = \inf_{m \ge 1} f_m + \] $(\mathscr{A}, \overline{\mathscr{B}})$-messbar nach (1)(ii). + + Definiere nun $h_m \coloneqq \inf_{n \ge m} X_n$. $h_m$ messbar nach (1) (ii) + $\forall m \in \N$. Dann ist auch + \[ + \liminf_{n \to \infty} X_n = \sup_{m \ge 1} \inf_{n \ge m} X_n = \sup_{m \ge 1} h_m + \] $(\mathscr{A}, \overline{\mathscr{B}})$-messbar nach (1)(i). + \end{proof} + \item Beh.: $\lim_{n \to \infty} X_n$ $(\mathscr{A}, \overline{\mathscr{B}})$-messbar. + \begin{proof} + Sei $X = \lim_{n \to \infty} X_n$. Dann ist + $X = \liminf_{n \to \infty} X_n = \limsup_{n \to \infty} X_n$, also + $X$ $(\mathscr{A}, \overline{\mathscr{B}})$-messbar nach (2). + \end{proof} + \end{enumerate} + \item Beh.: $Y$ $(\mathscr{A}, \mathscr{B})$-messbar. + \begin{proof} + Beachte, dass es für $\mathscr{B}$ genügt, die offenen Intervalle + $(a, \infty)$ für $a \in \R$ zu betrachten. + + Sei $a \in \R$ bel. + \begin{itemize} + \item Falls $a \le 0$, dann ist $0, 1 \in (a, \infty)$, also + $Y^{-1}((a, \infty)) = \Omega \in \mathscr{A}$. + \item Falls $0 < a < 1$: Dann ist $1 \in (a, \infty)$ und $0 \not\in (a, \infty)$. Damit + folgt + \begin{salign*} + Y^{-1}((a, \infty)) &= \{ \omega \in \Omega \mid X_1(\omega) > X_2(\omega) \} \\ + &= \{ \omega \in \Omega \mid X_1(\omega) - X_2(\omega) > 0\} \\ + &= \{ \omega \in \Omega \mid (X_1 - X_2)(\omega) \in (0, \infty]\} \\ + &= (X_1 - X_2)^{-1}((0, \infty]) + .\end{salign*} + Da $X_1, X_2$ $(\mathscr{A}, \overline{\mathscr{B}})$-messbar, ist + nach VL auch $X_1 - X_2$ $(\mathscr{A}, \overline{\mathscr{B}})$ messbar. + Da weiter $(0, \infty] \in \overline{\mathscr{B}}$, folgt also + $(X_1 - X_2)^{-1}((0, \infty]) \in \mathscr{A}$. + \item Falls $a \ge 1$, dann ist $0, 1 \not\in (a, \infty)$, also + $Y^{-1}((a, \infty)) = \infty \in \mathscr{A}$. + \end{itemize} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}