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\begin{enumerate}[a)] + \item Beh.: $\forall \epsilon > 0$ ex. $K \subseteq \Omega$ kompakt, s.d. + $\mathscr{L}^{1}(\Omega \setminus K) < \epsilon$ und $f|_K$ stetig. + \begin{proof} + Zunächst ist $f \in L^{\infty}(\Omega)$, also + ist $\text{ess sup}_{\Omega} |f| < \infty$. Damit folgt + \begin{salign*} + \int_{\Omega}^{} |f| \d{\mathscr{L}^{1}} + \le \int_{\Omega}^{} \text{ess sup}_{\Omega} |f| \d{\mathscr{L}^{1}} + = \text{ess sup}_{\Omega}|f| \cdot \mathscr{L}^{1}(\Omega) < \infty + .\end{salign*} + Also folgt $f \in L^{1}(\Omega)$. Dann ex. nach Hinweis eine Folge + $f_k \in C^{0}(\Omega) \cap L^{1}(\Omega)$ mit $f_k \to f$ in $L^{1}(\Omega)$. Nach + 3.36 existiert dann eine Teilfolge $f_{k_j} \in C^{0}(\Omega) \cap L^{1}(\Omega)$ + mit $f_{k_j} \xrightarrow{j \to \infty} f$ f.ü.. + + + Sei nun $\epsilon > 0$. Dann existiert nach Satz von Egorov + ein $E_{\epsilon} \in \mathscr{B}(\Omega)$ mit + $\mathscr{L}^{1}(E_\epsilon) \le \frac{\epsilon}{8}$ (i) + und $f_{k_j} \rightrightarrows f$ in $\Omega \setminus E_{\epsilon}$. + + Da $\mathscr{L}^{1}$ regulär ex. ein $\tilde{U} \supset E_{\epsilon}$ mit + $\tilde{U}$ offen, s.d. + $\mathscr{L}^{1}(\tilde{U}) < \mathscr{L}^{1}(E_{\epsilon}) + \frac{\epsilon}{8}$ (ii). + Setze nun $U \coloneqq \tilde{U} \cap \Omega$. Es ist $U$ offen, + da $\tilde{U}$ offen und $\Omega$ offen. Außerdem ist $E_{\epsilon} \subseteq U$, + da $E_{\epsilon} \subseteq \Omega$ und $E_{\epsilon} \subseteq \tilde{U}$. + Weiter ist + $U \subseteq \tilde{U}$ und damit folgt mit Monotonie von $\mathscr{L}^{1}$: + \begin{salign*} + \mathscr{L}^{1}(U) \le + \mathscr{L}^{1}(\tilde{U}) + \stackrel{\text{(ii)}}{<} \mathscr{L}^{1}(E_{\epsilon}) + \frac{\epsilon}{8} + \stackrel{\text{(i)}}{\le } \frac{\epsilon}{8} + \frac{\epsilon}{8} = \frac{\epsilon}{4} + \quad \text{(iii)} + .\end{salign*} + Weiter ex., da $\mathscr{L}^{1}$ regulär, ein $V \subseteq \Omega$ mit + $V$ abgeschlossen, s.d. + $\mathscr{L}^{1}(V) > \mathscr{L}^1(\Omega) - \frac{\epsilon}{4}$ (iv). + + Betrachte nun $\tilde{K} \coloneqq U^{c} \cap V$. Da $U$ offen, ist $U^{c}$ abgeschlossen + und da $V$ abgeschlossen ist auch $\tilde{K}$ abgeschlossen. Damit ist $\forall x \in \tilde{K}$ + folgt $x \not\in U$. Da $E_{\epsilon} \subseteq U$ folgt + $x \not\in E_{\epsilon}$. Also ist $x \in \Omega \setminus E_{\epsilon}$. Es folgt + also $\tilde{K} \subseteq \Omega \setminus E_{\epsilon}$ und damit + $f_{k_j} \rightrightarrows f$ in $\tilde{K}$. + + Es ist $\tilde{K} \subseteq V \subseteq \Omega$ und + damit mit Monotonie $\mathscr{L}^{1}(\tilde{K}) \le \mathscr{L}^{1}(\Omega) < \infty$. Damit gilt + \begin{salign*} + \mathscr{L}^{1}(\Omega \setminus \tilde{K}) &= + \mathscr{L}^{1}(\Omega) - \mathscr{L}^{1}(\tilde{K}) \\ + &= \mathscr{L}^{1}(\Omega) - \mathscr{L}^{1}(V \setminus (U \cap V)) \\ + &\stackrel{\text{(iv)}}{<} \mathscr{L}^{1}(\Omega) - \mathscr{L}^{1}(\Omega) + + \frac{\epsilon}{4} + \mathscr{L}^{1}(U \cap V) \\ + &\stackrel{\text{Monotonie}}{\le } \frac{\epsilon}{4} + \mathscr{L}^{1}(U) \\ + &\stackrel{\text{(iii)}}{\le } \frac{\epsilon}{4} + \frac{\epsilon}{4} \\ + &= \frac{\epsilon}{2} \qquad \text{(v)} + .\end{salign*} + + Betrachte nun $\tilde{K}_n \coloneqq \tilde{K} \cap [-n, n]$. Dann + ist $\tilde{K}_n \nearrow \tilde{K}$ und damit + $\exists n_0 \in \N$, s.d. + $| \mathscr{L}^{1}(\tilde{K}) - \mathscr{L}^{1}(\tilde{K}_{n_0})| < \frac{\epsilon}{2}$. + Setze $K \coloneqq \tilde{K}_{n_0}$. Da $K \subseteq \tilde{K}$ folgt + $\mathscr{L}^{1}(K) \le \mathscr{L}^{1}(\tilde{K})$ und damit insbesondere + $\mathscr{L}^{1}(K) > \mathscr{L}^{1}(\tilde{K}) - \frac{\epsilon}{2}$ (vi). Dann gilt + \begin{salign*} + \mathscr{L}^{1}(\Omega \setminus K) = \mathscr{L}^{1}(\Omega) + - \mathscr{L}^{1}(K) \stackrel{\text{(vi)}}{<} \mathscr{L}^{1}(\Omega) + - \mathscr{L}^{1}(\tilde{K}) + \frac{\epsilon}{2} + = \mathscr{L}^{1}(\Omega \setminus \tilde{K}) + \frac{\epsilon}{2} + \stackrel{\text{(v)}}{<} \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon + .\end{salign*} + Es ist $K$ beschränkt, da $K \subseteq [-n_0, n_0]$ und $K$ abgeschlossen, + da $\tilde{K}$ und $[-n_0, n_0]$ abgeschlossen. Also ist $K$ kompakt. Außerdem + ist $f_{k_j} \rightrightarrows f$ in $\tilde{K} \supseteq K$. Da + $f_{k_j} \in C^{0}(\Omega)$ ist also $f|_K$ stetig. + \end{proof} + \item Sei $\epsilon > 0$ beliebig. Dann sei $(q_i)_{i \in \N}$ eine Abzählung von $\Q$. + Dann betrachte + \[ + U \coloneqq \bigcup_{i \in I} \left( q_i - \frac{\epsilon}{2^{i+2}}, q_i + \frac{\epsilon}{2^{i+2}} \right) + .\] Dann ist $U$ offen als Vereinigung offener Mengen und $U^{c}$ abgeschlossen. Setze + nun $K \coloneqq U^{c} \cap \left[ \frac{\epsilon}{4}, 1 - \frac{\epsilon}{4} \right] \subseteq (0,1)$. + Dann ist $K$ abgeschlossen als Schnitt abgeschlossener Mengen. + Da $\left[ \frac{\epsilon}{4}, 1 - \frac{\epsilon}{4} \right] $ beschränkt, ist auch $K$ beschränkt. + Also $K$ abgeschlossen und beschränkt, damit kompakt. + + Weiter ist $\Q \subseteq U$, d.h. $\Q \cap K \subseteq \Q \cap U^{c} = \emptyset$. Das + heißt $\forall x \in K\colon f(x) = 0$. Also insbesondere $f|_K \equiv 0$ und + damit stetig. + + Außerdem folgt, da $\mathscr{L}^{1}((0,1)) = 1$, insbesondere endlich: + \begin{salign*} + \mathscr{L}^{1}((0,1) \setminus K) &= 1 - \mathscr{L}^{1}(K) \\ + &= 1 - \mathscr{L}^{1}\left(U^{c} \cap \left[ \frac{\epsilon}{4}, 1 - \frac{\epsilon}{4} \right] \right) \\ + &= 1 - \mathscr{L}^{1}\left( \left[ \frac{\epsilon}{4}, 1 - \frac{\epsilon}{4} \right] + \setminus \left( U \cap \left[ \frac{\epsilon}{4}, 1 - \frac{\epsilon}{4} \right] \right) \right) \\ + &\stackrel{\text{Monotonie}}{\le} 1 - \mathscr{L}^{1}\left( \left[ \frac{\epsilon}{4}, 1- \frac{\epsilon}{4} \right] \right) + + \mathscr{L}^{1}(U) \\ + &\stackrel{\text{Subadd}}{\le} \frac{\epsilon}{2} + \sum_{i\in \N} \frac{\epsilon}{2^{i+1}} \\ + &= \frac{\epsilon}{2} + \frac{\epsilon}{2} \\ + &= \epsilon + .\end{salign*} + Also liegt kein Widerspruch zu (a) vor. + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Definiere $f\colon \R \to \R$, $f(x) = 0$ $\forall x \in \R$. + + Beh.: $f_k \to f$ punktweise. + \begin{proof} + Sei $\epsilon > 0$ und $x \in \R$. Zunächst gilt + $\forall k \in \N\colon I_k \subseteq (0, \infty)$. + + Falls $x \le 0$ ist $f_k(x) = 0$, da $\forall k \in \N\colon x \not\in I_k$. + Falls nun $x > 0$. Dann ex. ein $n \in \N$, s.d. $2^{-n} < x$. Dann gilt + $\forall k \ge n\colon$ $2^{-k} \le 2^{-n}$ und damit + $ I_k \subseteq (0, 2^{-k}] \subseteq (0, 2^{-n}]$. Da $x > 2^{-n}$ folgt + also $x \not\in I_k$ und damit $f_k(x) = 0$, also insbesondere + $|f_k(x) - f(x)| = |f_k(x)| = 0 < \epsilon$. + \end{proof} + + Beh.: $f_k$ konvergiert nicht gleichmäßig. + \begin{proof} + Setze $\epsilon \coloneqq \frac{\sqrt{2} }{2}$. Sei nun $k \in \N$. Dann + setze $x \coloneqq 2^{-k}$. Dann ist $x \in I_k$ und damit + \[ + f_k(x) = \frac{1}{\sqrt{2^{-k}} } \ge \frac{1}{\sqrt{2^{-1}} } = \sqrt{2} > \epsilon + .\] + \end{proof} + + Beh.: $f_k \to f$ im Maß. + \begin{proof} + Es ist $\forall x \in [0,1]^{c}\colon f_k(x) = 0$, denn + $\forall k \in \N\colon I_k \subseteq [0, 2^{-k}] \subseteq [0,2^{-1}] \subseteq [0,1]$. + Das heißt $\forall x \in \R \setminus [0,1]\colon |f_k(x) - f(x)| = 0$. + + D.h. es genügt $f_k$ auf $X \coloneqq [0,1]$ zu betrachten. Dann gilt jedoch + $\mathscr{L}^{1}(X) = 1 < \infty$ und damit mit 3.35, da $f_k \to f$ punktweise, + konvergiert $f_k \to f$ im Maß auf $X$. Insgesamt folgt also die Behauptung. + \end{proof} + \item Beh.: Für $1 \le p < \infty$ gilt: $f_k$ konvergiert in $L^{p}(\R)$ $\iff$ $p < 2$. + \begin{proof} + Zunächst ist für $k \in \N$ $\frac{1}{\sqrt{x} }$ stetig und damit auf $I_k$ $f_k$ stetig + und damit R.-integrierbar, also + auch Lebesgue-integrabel und die Werte stimmen überein. Es kann also auf $I_k$ der HDI + angewandt werden. + Da weiter $\chi_{I_k}$ messbar, ist $f_k$ als Produkt messbarer Funktionen messbar. + + Sei $1 \le p < \infty$ beliebig. + \begin{itemize} + \item Falls $p = 2$: Dann sei $k \in \N$ bel. Mit der Vorüberlegung folgt direkt + \begin{salign*} + \Vert f_k - f \Vert_{2}^{2} = \int_{I_k}^{} \frac{1}{x} \d{x} + = \log(|x|) \Big|_{2^{-(k+1)}}^{2^{-k}} = - k \log(2) + (k+1)\log(2) = \log(2) + \xrightarrow{k \to \infty} \log(2) + .\end{salign*} + Also $f_k$ konvergiert nicht in $L^{2}(\R)$. + \item Falls $p \neq 2$: Dann sei $k \in \N$. Dann gilt + mit $\mu \coloneqq \frac{2}{2-p} \left( 1 - 2^{-\frac{2-p}{2}} \right) $: + \begin{salign*} + \Vert f_k \Vert_{p}^{p} = \int_{\R}^{} |f_k|^{p} \d{x} + = \int_{I_k}^{} x^{-\frac{p}{2}} \d{x} + = \frac{2}{2-p} 2^{-k \frac{2-p}{2}} \left( 1 - 2^{- \frac{2-p}{2}} \right) + = \mu 2^{-k \frac{2 - p}{2}} + .\end{salign*} + Falls $p < 2$ ist $\frac{2-a}{2} > 0$ und damit + $\Vert f_k \Vert_p^{p} = \mu 2^{- k \frac{2-p}{2}} \xrightarrow{k \to \infty} 0$. + Also konvergiert $f_k$ in $L^{p}(\R)$ für $p < 2$. + + Falls $p > 2$, dann ist $\frac{2 - p}{2} < 0$ und + damit $ - k \frac{2-p}{2} \xrightarrow{k \to \infty} \infty$, also + $\Vert f_k \Vert_{p}^{p} = \mu 2^{-k \frac{2-p}{2}} \xrightarrow{k \to \infty} \infty$. + Also konvergiert $f_k$ in $L^{p}(\R)$ für $p > 2$ nicht. + \end{itemize} + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Beh.: $\mathscr{B}(\R^{n}) = \mathscr{B}(\R)^{n}$. + \begin{proof} + \begin{itemize} + \item ,,$\subseteq $'': Betrachte + $\mathscr{O} \coloneqq \{A \subseteq \R^{n} \mid A \text{ offen}\} $. Dann + ist $\sigma(\mathscr{O}) = \mathscr{B}(\R^{n})$. Betrachte weiter + $\mathscr{R} \coloneqq \{ \bigtimes_{i=1}^{n} (a_i, b_i) \mid + a_i, b_i \in \Q\} $. Dann ist + $\mathscr{R} \subseteq \mathscr{B}(\R)^{n}$, denn $\forall a, b \in \Q$ ist + $(a,b)$ offen und damit $(a,b) \in \mathscr{B}(\R)$. + + Sei $\Omega \in \mathscr{O}$. + Da $\Q$ dicht + in $\R$, ex. $R_i \in \mathscr{R}$, s.d. + $\Omega = \bigcup_{i \in \N} R_i$. Damit folgt + $\Omega \in \sigma(\mathscr{R})$ und damit + $\mathscr{O} \subseteq \sigma(\mathscr{R})\subseteq \mathscr{B}(\R)^{n}$. + Da $\sigma(\mathscr{O}) = \mathscr{B}(\R^{n})$, folgt + $\mathscr{B}(\R^{n}) \subseteq \mathscr{B}(\R)^{n}$. + \item ,,$\supseteq$'': Dazu betrachte + $\mathscr{K} = \{ \bigtimes_{i=1}^{n} A_i \mid A_i \in \mathscr{B}(\R)\} $. + Dann ist nach Definition der Produktalgebra $\sigma(\mathscr{K}) = \mathscr{B}(\R)^{n}$. + + Betrachte nun $\pi_i \colon \R^{n} \to \R$ mit + $(x_1, \ldots, x_i, \ldots, x_n) \mapsto x_i$. $\pi_i$ ist stetig und damit + insbesondere $(\mathscr{B}(\R^{n}), \mathscr{B}(\R))$-messbar. + Seien nun $A_i \in \mathscr{B}(\R)$ für $i = 1, \ldots, n$. Dann ist + \begin{salign*} + \bigtimes_{i=1}^{n} A_i = \bigcap_{i=1}^{n} (A_i \cap \R) + = \bigcap_{i=1}^{n} \underbrace{\pi_{i}^{-1}(A_i)}_{\in \mathscr{B}(\R^{n})} + \in \mathscr{B}(\R^{n}) + .\end{salign*} + Also ist $\mathscr{K} \subseteq \mathscr{B}(\R^{n})$ und damit + $\mathscr{B}(\R)^{n} = \sigma(\mathscr{K}) \subseteq \mathscr{B}(\R^{n})$. + \end{itemize} + \end{proof} + \item Beh.: $N \times \Omega \in \mathscr{B}(\R^{n})$. + \begin{proof} + Es ist $N \in \mathscr{B}(\R)$ und $\Omega \in \mathscr{B}(\R^{n-1})$. Dann gilt + nach Definition + \[ + N \times \Omega \in \mathscr{B}(\R) \times \mathscr{B}(\R^{n-1}) + \stackrel{\text{(a)}}{=} \mathscr{B}(\R) \times \mathscr{B}(\R)^{n-1} + = \mathscr{B}(\R)^{n} \stackrel{\text{(a)}}{=} \mathscr{B}(\R^{n}) + .\] + \end{proof} + Beh.: $\mathscr{L}^{n}(N \times \Omega) = 0$. + \begin{proof} + Es ist nach Definition + $\mathscr{L}^{n} = (\mathscr{L}^{1})^{n} = \mathscr{L}^{1} \times (\mathscr{L}^{1})^{n-1}$. + Es ist $\mathscr{L}^{n}$ das Produktmaß von $\mathscr{L}^{1}$ und $(\mathscr{L}^{1})^{n-1}$. + auf den $\sigma$-endlichen Maßräumen $(\R, \mathscr{B}(\R), \mathscr{L}^{1})$ und + $(\R^{n-1}, \mathscr{B}(\R^{n-1}), (\mathscr{L}^{1})^{n-1})$. Damit + gilt nach 4.5 und unserer Konvention $0 \cdot \infty = 0$: + \[ + \mathscr{L}^{n}(N \times \Omega) = \mathscr{L}^{1}(N) (\mathscr{L}^{1})^{n-1}(\Omega) + = 0 + .\] + \end{proof} + \item Beh.: Für $n \ge 2$ ist $(\mathscr{L}_1)^{n} \subsetneqq \mathscr{L}_n$. + \begin{proof} + Sei $V \subseteq [0,1]$ eine nicht lebesgue messbare Menge (solch eine Menge existiert + nach Vitali). Dann definiere $A \coloneqq V \times \{0\}^{n-1}$ und + $B \coloneqq \R \times \{0\}^{n-1}$. Dann ist $A \subseteq B$ + und es ist, da $\mathscr{L}^{n} = (\mathscr{L}_1)^{n}$, + der Konvention $\mathscr{L}^{0}(\{0\}^{0}) = 1$ und erneut $0 \cdot \infty = 0$: + \[ + \mathscr{L}^{n}(B) + = \mathscr{L}^{1}(\R) \underbrace{\mathscr{L}^{1}(\{0\})}_{= 0} + \mathscr{L}^{n-2}(\{0\}^{n-2}) + = 0 + .\] Da $A \subseteq B$ und $B$ $\mathscr{L}^{n}$-Nullmenge und + $(\R^{n}, \mathscr{L}_{n}, \mathscr{L}^{n})$ vollständiger Maßraum folgt + $A \in \mathscr{L}_n$. + + Ang.: $A \in (\mathscr{L}_1)^{n}$. Dann ist + $A \in \mathscr{L}_1 \times (\mathscr{L}_1)^{n-1}$. Mit $y = (0, \ldots, 0) \in \R^{n-1}$ + folgt, da $(\R, \mathscr{L}_1, \mathscr{L}^{1})$ + und $(\R^{n-1}, (\mathscr{L}_1)^{n-1}, (\mathscr{L}^1)^{n-1})$ $\sigma$-endliche + Maßräume: + \[ + \mathscr{L}_{1} \ni A^{y} = \{ x \in \R \mid (x, y) \in A\} + = \{ x \in \R \mid (x, 0, \ldots, 0) \in A\} + = V + .\] Also $V \in \mathscr{L}_1$ $\contr$. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[a)] + \item Beh.: Die Aussage ist falsch. + \begin{proof} + Betrachte $f_k \coloneqq \chi_{[k, k+1]}$. Dann ist $f_k \to f \equiv 0$ punktweise f.ü. + und $f_k, f \in L^{1}(\R)$, aber es gilt $\forall k \in \N$ + \[ + \int_{\R}^{} f_k \d{x} = \int_{[k, k+1]}^{} \d{x} = \mathscr{L}^{1}([k, k+1]) + = 1 \neq 0 = \int_{\R}^{} f \d{x} + .\] + \end{proof} + \item Beh.: Die Aussage ist falsch. + \begin{proof} + Betrachte $f_k \coloneqq \chi_{[k, \infty)} \ge 0$. + Dann ist $f_k \searrow f \equiv 0$ punktweise f.ü. + und $f \equiv 0 \in L^{1}(\R)$, aber $\forall k \in \N$: + \[ + \int_{\R}^{} f_k \d{x} = \int_{[k, \infty)}^{} \d{x} = \infty \neq 0 = \int_{\R}^{} f \d{x} + .\] + \end{proof} + \item Beh.: Die Aussage stimmt. + \begin{proof} + Da $f_k \searrow f$ punktweise f.ü. gilt $f_k \le f_1$ $\forall k \in \N$ und + wegen $f_1 \in L^{1}$ und $f_1 \ge 0$ ist $|f_1| = f_1$ integrabel, also folgt + mit dominierter Konvergenz, dass $f$ integrabel und + \[ + \lim_{k \to \infty} \int_{\R}^{} f_k \d{x} = \int_{\R}^{} f \d{x} + .\] Da $f_k \ge 0$ $\forall k \in \N$ folgt $f \ge 0$. Da $f$ integrabel + also auch $|f| = f$ integrabel und damit $f \in L^{1}(\R)$. + \end{proof} + \item Beh.: Die Aussage stimmt nicht. + \begin{proof} + Definiere + \[ + J_{kj} = [j 2^{-k}, (j+1)2^{-k}] + \] für $k \in \N_0$ und $j \in \N_0$ mit $j+1 \le 2^{k}$. + + Betrachte $a_n \coloneqq \left\lfloor \log_2(n) \right\rfloor$ und + $b_n \coloneqq n \text{ mod } 2^{a_n}$. Damit definiere + \[ + I_n \coloneqq J_{a_n b_n} + \] und setze $f_k \coloneqq \chi_{I_k}$. Es ist $f_k \to f \equiv 0$ in $L^{1}(\R)$, denn + $\forall k \in \N$ gilt + \[ + \int_{\R}^{} |f_k - 0| \d{x} = \int_{I_k}^{} \d{x} = \mathscr{L}^{1}(I_k) + = 2^{-a_k} = 2^{- \left\lfloor \log_2(k) \right\rfloor} + \xrightarrow{k \to \infty} 0 + ,\] da $\log_2(k)$, insbesondere $\left\lfloor \log_2(k) \right\rfloor$, + monoton wachsend und unbeschränkt. + + Aber setze nun $\epsilon \coloneqq \frac{1}{2}$. Dann sei + $x \in [0,1]$ und $k \in \N$ beliebig. + Dann setze + $n \coloneqq 2^{a_k + 1} = 2^{\left\lfloor \log_2(k) \right\rfloor + 1}$. Dann gilt + $n \ge k$. Da $x \in [0,1]$ ex. insbesondere ein $j \in \N_0$ mit + $j+1 \le 2^{n}$, s.d. + $x \in J_{nj}$. Dann setze $n_0 \coloneqq n + j$. Dann ist + $a_{n_0} = a_n$, da $j < 2^{n}$ und $b_{n_0} = n_0 \text{ mod }2^{a_{n}} = j$. Also + $I_{n_0} = J_{nj}$. Und damit $x \in I_{n_0}$. Außerdem ist $n_0 \ge k$. Damit folgt + dann + \[ + |f_{n_0}(x) - f(x)| = |f_{n_0}(x)| = \chi_{I_{n_0}}(x) = 1 > \frac{1}{2} = \epsilon + .\] Also konvergiert $f_k$ für kein $x \in [0,1]$ punktweise gegen $f \equiv 0$ und + da $\mathscr{L}^{1}([0,1]) > 0$ konvergiert $f_k$ nicht f.ü. punktweise. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\end{document}