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zyoSfHI!oGo$Q?_;DxMFhF(-gdo0=ge#h`2Qk%ucVrlh+VL3P*%>Ac0?;)U-niXi-qnd()!@yYoM)Q!hiG2WIdyqh-| z9gAO$l0DwMUIl@S&h=<(tyMXkp@op;e8+$VKim9`xq~ED^SUE08EVfz`(wD5f8Fb% zxO5JbZs7YLj7&o#r6P^|zW{lIxK988 literal 0 HcmV?d00001 diff --git a/ws2020/ana/uebungen/ana4.tex b/ws2020/ana/uebungen/ana4.tex new file mode 100644 index 0000000..cdbec06 --- /dev/null +++ b/ws2020/ana/uebungen/ana4.tex @@ -0,0 +1,288 @@ +\documentclass[uebung]{../../../lecture} + +\title{Analysis 3: Übungsblatt 4} +\author{Leon Burgard, Christian Merten} +\usepackage[]{mathrsfs} + +\begin{document} + +\punkte + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Beh.: $f$ messbar und + \[ + \lim_{k \to \infty} \int_{X}^{} f_k \d{\mu} + = \int_{X}^{} f \d \mu + .\] + \begin{proof} + Es ist $f_k$ messbar $\forall k \in \N$ und $\lim_{k \to \infty} f_k = f$, also ist + $f$ nach VL messbar. + + Zunächst gilt $\forall k \in \N$, da $f \ge f_k \ge g$ auch + $- f_- \ge - f_{k_-} \ge - g_-$, also insbesondere + $f_- \le f_{k_-} \le g_-$. Da $g$ integrierbar folgt damit mit + der Monotonie des Integrals für nicht-negative Funktionen + \[ + \int_{X}^{} f_- \d \mu \le \int_{X}^{} f_{k_-} \d \mu \le \int_{X}^{} g_- \d \mu < \infty + \quad (*) + .\] + Insbesondere ist $\int_{X}^{} f \d \mu$ und $\int_{X}^{} f_k \d \mu$ endlich oder + $+ \infty$. + \begin{itemize} + \item Falls $\int_{X}^{} f \d \mu = \infty$, dann ist bereits + $\int_{X}^{} f_+ \d \mu = \infty$, denn nach $(*)$ + ist $\int_{X}^{} f_- \d \mu < \infty$. Da + $f_k \nearrow f$ folgt auch $f_{k_+} \nearrow f_+$. Da + $f_{k_+}$ außerdem nicht-negativ und Folge messbarer Funktionen, folgt + \[ + \lim_{k \to \infty} \int_{X}^{} f_{k_+} \d \mu + \stackrel{\text{3.17}}{=} \int_{X}^{} \lim_{k \to \infty} f_{k_+} \d \mu + = \int_{X}^{} f_+ \d \mu = \infty + .\] + Wegen $(*)$ ist $f_{k_-} < \infty$ $\forall k \in \N$. Damit folgt + insgesamt $\lim_{k \to \infty} \int_{X} f_{k} \d \mu = \int_{X}^{} f \d \mu $. + \item Falls $\exists n \in \N$, s.d. $\int_{X}^{} f_n \d \mu = \infty$, dann ist + wegen $(*)$ wieder $\int_{X}^{} f_{n_+} \d \mu = \infty$ und wegen + $f_{k_+} \nearrow f_+$ und der Monotonie des Integrals für nicht-negative + Funktionen gilt dies auch $\forall k \ge n$. Damit folgt ganz analog, da + $f_{k_+}$ nicht-negativ und messbar: + \[ + \int_{X}^{} f_+ \d \mu + = \int_{X}^{} \lim_{k \to \infty} f_{k_+} \d \mu + \stackrel{\text{3.17}}{=} + \lim_{k \to \infty} \int_{X}^{} f_{k_+} \d \mu + = \infty + .\] Wegen $(*)$ ist $f_{-} < \infty$. Damit folgt insgesamt + $\int_{X}^{} f \d \mu = \lim_{k \to \infty} \int_{X}^{} f_k \d \mu $. + \item Falls nun $f$ und $f_k$ integrierbar $\forall k \in \N$ betrachte: + $h_n \coloneqq f_n - g \ge 0$. Dann ist $h_n \nearrow (f - g)$. Da $f_n$ und + $g$ integrierbar, ist auch $h_n$ integrierbar $\forall n \in \N$. Damit folgt + \begin{salign*} + \lim_{k \to \infty} \int_{X}^{} f_k \d \mu + - \underbrace{\int_{X}^{} g \d \mu}_{< \infty} + &= \lim_{k \to \infty} \left[ \int_{X}^{} f_k \d \mu - \int_{X}^{} g \d \mu \right] \\ + &\stackrel{f_n, g \text{ integrabel}}{=} + \lim_{k \to \infty} \int_{X}^{} h_{k} \d \mu \\ + &\stackrel{\text{3.17}}{=} + \int_{X}^{} (f-g) \d \mu \\ + &\stackrel{f, g \text{ integrabel}}{=} + \int_{X}^{} f \d \mu - \underbrace{\int_{X}^{} g \d \mu}_{< \infty} + \intertext{Damit folgt} + \lim_{k \to \infty} \int_{X}^{} f_k \d \mu &= \int_{X}^{} f \d \mu + .\end{salign*} + \end{itemize} + \end{proof} + \item Beh.: + \[ + \int_{X}^{} \liminf_{k \to \infty} f_k \d \mu \le \liminf_{k \to \infty} \int_{X}^{} f_k \d \mu + .\] + \begin{proof} + Betrachte $f \coloneqq \liminf_{n \to \infty} f_n$ und + $g_n \coloneqq \inf \{ f_k \}_{k=n}^{\infty} \le f_n$. Nach Definition von + $\liminf$ folgt $g_n \nearrow f$. Da $g \le f_k$ $\forall k \in \N$ folgt insbesondere + \[ + g \le \inf_{k \in \N} \{ f_k \} \le \inf \{ f_k \}_{k=n}^{\infty} = g_n \qquad \forall n \in \N + .\] Damit sind die Bedingungen für (a) erfüllt, also folgt + \begin{salign*} + \int_{X}^{} \liminf_{n \to \infty} f_n \d \mu + = \int_{X}^{} f \d \mu + \stackrel{\text{(a)}}{=} \lim_{n \to \infty} \int_{X}^{} g_n \d \mu + \le \liminf_{n \to \infty} \int_{X}^{} f_n \d \mu + .\end{salign*} + \end{proof} + \item Beh.: Ohne die Voraussetzung sind die Aussagen i.A. falsch. + \begin{proof} + Im Allgemeinen sind die Integrale nicht mal wohldefiniert, da für + nicht-negative Funktionen das Integral + \[ + \int_{}^{} f \d \mu \coloneqq \int_{}^{} f_+ \d \mu - \int_{}^{} f_- \d \mu + \] gesetzt wird. Falls also $\int_{}^{} f_- \d \mu = \int_{}^{} f_+ \d \mu = \infty$, + ist dieser Ausdruck nicht wohldefiniert. + + Unter wohlwollender Interpretation der Definition im Skript, wenn nur + entweder $\int_{}^{} f_+ \d \mu = \infty$ oder $\int_{}^{} f_- \d \mu = \infty$, + kann jedoch auch folgender Fall betrachtet werden: + $X = \N$, $\mathcal{E} = \mathscr{P}(\N)$ und $\mu$ Zählmaß. Dann + definiere + \[ + f_k(n) \coloneqq \begin{cases} + -1 & n \ge k \\ + 0 & \text{sonst} + \end{cases} + .\] Dann folgt direkt $f_k \nearrow 0 =: f$ für $k \to \infty$. Aber + es ist $\int_{}^{} f_k \d \mu = - \infty$ $\forall k \in \N$, also insbesondere + \[ + \lim_{k \to \infty} \int_{X}^{} f_k \d \mu = -\infty \neq 0 = \int_{X}^{} \lim_{k \to \infty} f_k \d \mu + .\] Es ist weiter $\liminf_{n \to \infty} f_n = \lim_{n \to \infty} f_n = 0$. Damit folgt + \[ + \int_{X}^{} \liminf_{k \to \infty} f_k \d \mu = 0 > -\infty + = \liminf_{k \to \infty} - \infty = \liminf_{k \to \infty} \int_{X}^{} f_k \d \mu + .\] + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Beh.: + \[ + \lim_{k \to \infty} \int_{0}^{\infty} e^{-x} \cos\left( \frac{x}{k} \right) \d{\lambda(x)} + = 1 + .\] + \begin{proof} + Zunächst ist $e^{-x} \cos\left( \frac{x}{k} \right) $ stetig, insbesondere + R.-integrierbar. Es kann also der HDI angewendet werden. Damit folgt + mit der stetig differenzierbaren Substitution $y = \frac{x}{k}$: + \begin{salign*} + \int_{0}^{\infty} e^{-x} \cos\left( \frac{x}{k} \right) \d{\lambda(x)} + &= \int_{0}^{\infty} k e^{-ky} \cos(y) \d{\lambda(y)} \\ + &\stackrel{\text{part. Integr.}}{=} + -e^{-ky} \cos(y) \Big|_{0}^{\infty} - \int_{0}^{\infty} e^{-ky}\sin(y) \d{\lambda(y)} \\ + &= 1 - \int_{0}^{\infty} e^{-ky} \sin(y) \d{\lambda(y)} \qquad (*) + .\end{salign*} + Betrachte nun $f_k(y) \coloneqq e^{-ky} \sin(y)$. Falls $y = 0$: ist + $f_k(0) = e^{0} \sin(0) = 0$ $\forall k \in \N$. Falls $y > 0$ folgt + $\lim_{k \to \infty} f_k(y) = \sin(y) \lim_{k \to \infty} e^{-ky} = 0$. Weiter + ist $|f_k(y)| = |\sin(y) e^{-ky}| \le |e^{-ky}| \le e^{-y}$. Außerdem + $e^{-y}$ integrabel, denn es ist $|e^{-y}|$ uneigentlich R.-integrabel, denn + \[ + \int_{0}^{\infty} e^{-y} \d{\lambda(y)} + = \lim_{z \to \infty} \int_{0}^{z} e^{-y} \d{\lambda(y)} + = \lim_{z \to \infty} \left[ -e^{-z} + e^{0} \right] = 1 < \infty + .\] + Damit ist $f_k$ durch eine integrable Funktion beschränkt. Damit folgt mit 3.19 + \begin{salign*} + \lim_{k \to \infty} \int_{0}^{\infty} e^{-x} \cos\left( \frac{x}{k} \right) \d\lambda(x) + &\stackrel{(*)}{=} 1 - \lim_{k \to \infty} \int_{0}^{\infty} e^{-ky} \sin(y) \d{\lambda(y)}\\ + &\stackrel{3.17}{=} 1 - \int_{0}^{\infty} \lim_{k \to \infty} e^{-ky} \sin(y) \d{\lambda(y)}\\ \\ + &= 1 - \int_{0}^{\infty} 0 \d{\lambda(y)} \\ + &= 1 + .\end{salign*} + \end{proof} + \item + \item + \item Beh.: Für $z > 0$ + \[ + \lim_{k \to \infty} \int_{z}^{\infty} \frac{1}{1+k^2x^2} \d{\lambda(x)} = 0 + .\] + \begin{proof} + Es ist $\frac{1}{1+k^2x^2}$ stetig, insbesondere R.-integrierbar, also folgt mit HDI, + der stetig diff'baren Substitution $y = kx$ und partieller Integration: + \begin{salign*} + \int_{z}^{\infty} \frac{1}{1+k^2x^2} \d{\lambda(x)} + &= \int_{\frac{z}{k}}^{\infty} \frac{1}{k} \left[ \frac{1}{1+y^2} \right] \d{\lambda(y)} \\ + &= \frac{1}{k} \int_{\frac{z}{k}}^{\infty} \frac{1}{1+y^2} \d{\lambda(y)} \\ + &= \frac{1}{k} \int_{0}^{\infty} \frac{1}{1+y^2} \chi_{A_k}(y) \d{\lambda(y)} + .\end{salign*} + Mit $A_k \coloneqq \{ y \in [0, \infty) \mid y \ge \frac{z}{k}\} $. Es + ist $A_k \xrightarrow{k \to \infty} [0, \infty)$. Betrachte + dann $f_k \coloneqq \frac{1}{1+y^2} \chi_{A_k}(y) \xrightarrow{k \to \infty} \frac{1}{1+y^2}$. + Außerdem ist $\frac{1}{1+y^2} \ge 0$ $\forall y \in [0, \infty)$, also + $f_k$ monoton wachsend, also $f_k \nearrow f$. Damit folgt + \begin{salign*} + \lim_{k \to \infty} \int_{z}^{\infty} \frac{1}{1+k^2x^2} \d{\lambda(x)} + &\stackrel{(*)}{=} \lim_{k \to \infty} \frac{1}{k}\int_{0}^{\infty} f_k(y) \d{\lambda(y)} \\ + &= \lim_{k \to \infty} \frac{1}{k} + \cdot \lim_{k \to \infty} \int_{0}^{\infty} \lim_{k \to \infty} f_k \d{\lambda(y)} \\ + &\stackrel{\text{3.17}}{=} + \lim_{k \to \infty} \frac{1}{k} \cdot \frac{\pi}{2} \\ + &= 0 + .\end{salign*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Beh.: $x \mapsto \partial_x f(x,y)$ ist messbar $\forall y \in Y$. + \begin{proof} + Sei $y \in Y$. Betrachte die Folge $f_n \colon X \to \R$ mit + \[ + f_n(x) \coloneqq \frac{f\left(x + \frac{1}{n}, y\right) - f(x, y)}{\frac{1}{n}} \qquad + \forall x \in X + .\] Dann ist, da $f$ differenzierbar bezüglich $x$, + $f_n(x) \xrightarrow{n \to \infty} \partial_xf(x,y)$. $\forall n \in \N$ + ist $f_n$ messbar, da $f$ diff'bar insbesondere stetig bezüglich $x$ und $\frac{1}{n} \neq 0$, + also $f_n$ als Linearkombination messbarer Funktionen messbar. + + Damit ist $\partial_x f = \lim_{n \to \infty} f_n$ messbar. + \end{proof} + \item Beh.: $F$ diff'bar und + \[ + F'(x) = \int_{Y}^{} \partial_x f(x,y) \d{\mu(y)} + .\] + \begin{proof} + Sei $x \in X$ und $(h_n)_{n \in \N}$ Folge mit $h_n \xrightarrow{n \to \infty} 0$, s.d. + $x + h_n \in X$ $\forall n \in \N$. Betrachte dann + \[ + f_n(y) \coloneqq \frac{f(x + h_n, y) - f(x, y)}{h_n} + .\] Damit folgt: $\forall n \in \N$ existiert nach MWS der Differentialrechnung + ein $c \in (x, x + h_n) \subseteq X$, s.d. + \[ + f_n(y) = \partial_x f(c, y) + .\] Damit folgt insbesondere + \[ + |f_n(y)| \le \sup_{x \in X} |\partial_x f(x,y)| \le g(y) \qquad \forall n \in \N + .\] + Damit folgt insgesamt, da $|f_n(y)| \le g(y)$ und da $f$ diff'bar auch + $f_n \xrightarrow{n \to \infty} \partial_x f$. Außerdem $f_n$ messbar und + $g$ nach Voraussetzung integrabel. Damit ist nach 3.19 + $\partial_x f(x,y)$ integrabel bezüglich $y$ und + es folgt + \begin{salign*} + \lim_{n \to \infty} \frac{F(x+h_n, y) - F(x,y)}{h_n} + &\stackrel{\text{Def.}}{=} \lim_{n \to \infty} \int_{X}^{} + \frac{f(x+h_n, y) - f(x,y)}{h_n} \d \mu(y) \\ + &\stackrel{\text{3.19}}{=} \int_{X}^{} \lim_{n \to \infty} f_n \d \mu(y) \\ + &= \int_{X}^{} \partial_x f(x,y) \d \mu(y) + \intertext{ Da $(h_n)_{n \in \N}$ beliebig, ist also $F$ diff'bar und es folgt} + F'(x) &= \int_{X}^{} \partial_x f(x,y) \d \mu(y) + .\end{salign*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Beh.: Die Definition ist unabhängig von der Darstellung von $f$. + \begin{proof} + Sei $f = \sum_{k=1}^{K} \gamma_k \chi_{C_k}$ mit $K \in \N$ und + $C_i \cap C_j = \emptyset$ und $\gamma_i \neq \gamma_j$ für $i \neq j$. Diese Darstellung + existiert, da $f$ einfach. Sei nun weiter + $f = \sum_{k=1}^{N} \alpha_k \chi_{A_k}$. O.E. $\alpha_k \neq 0$ $\forall k=1,\ldots,N $ + (diese Summanden geben keinen Beitrag zum Integral). Dann für + $k \in \{1, \ldots, N\} $ und $j \in \{1, \ldots, K\} $ definiere + \[ + A_{kj} \coloneqq A_k \cap C_j + .\] Dann ist $A_k = \bigcupdot_{j=1}^{K} A_{kj}$ und $C_j = \bigcupdot_{k=1}^{N} A_{kj}$, da + die $C_j$ paarweise disjunkt. Weiter + gilt $\gamma_j \mu(C_j) = \sum_{k=1}^{N} \alpha_k \mu(A_{kj})$. Damit folgt + \begin{salign*} + \sum_{k=1}^{N} \alpha_k \mu(A_k) + &\stackrel{\mu \text{ Maß}}{=} \sum_{k=1}^{N} \alpha_k \sum_{k=1}^{K} \mu(A_{kj}) \\ + &= \sum_{j=1}^{K} \sum_{k=1}^{N} \alpha_k \mu(A_{kj}) \\ + &= \sum_{j=1}^{K} \gamma_j \mu(C_j) + .\end{salign*} + Damit ist die Definition unabhängig von der Darstellung. + \end{proof} + \item Beh.: Für $f \in \mathscr{S}_+$ stimmen beide Definitionen des Integrals überein. + \begin{proof} + Sei $f = \sum_{k=1}^{N} \alpha_k \chi_{A_k} \in \mathscr{S}_+$. Dann ist + $f \in \{ g \le f \mid g \in \mathscr{S}_+(X, \mathcal{E}, \mu\} $. Sei + nun $g \in \mathscr{S}_+(X, \mathcal{E}, \mu)$ mit $g \le f$. Dann ist + wegen der Monotonie des Integrals auf $\mathscr{S}_+$: + \[ + \int_{X}^{} g \d \mu \le \int_{X}^{} f \d \mu = \sum_{k=1}^{N} \alpha_k \mu(A_k) + .\] Damit folgt + \begin{salign*} + \sum_{k=1}^{N} \alpha_k \mu(A_k) + &= \int_{X}^{} f \d \mu \\ + &= \max \left\{ \int_{X}^{} g \d \mu \mid g \le f, g \in \mathscr{S}_+\right\} \\ + &= \sup \left\{ \int_{X}^{} g \d \mu \mid g \le f, g \in \mathscr{S}_+\right\} + .\end{salign*} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}