From 44e707b842edc91d41e80c8662bfe57fedf9ba40 Mon Sep 17 00:00:00 2001 From: Liz Date: Wed, 17 Feb 2021 18:53:08 +0100 Subject: [PATCH 1/6] Kapitel 4 VL 12 --- analysis_3_skript.pdf | Bin 626685 -> 628406 bytes chapter4.tex | 77 +++++++++++++++++++++++++++++++++++------- 2 files changed, 65 insertions(+), 12 deletions(-) diff --git a/analysis_3_skript.pdf b/analysis_3_skript.pdf index 2933882859c2a2dfcb381641d43e4f5c23f06983..f3287bb0156d91fbec58fbd903ddb9c4f836d0cb 100644 GIT binary patch delta 285658 zcmV(>K-j}r$5^mx{)mO>*BCit`H~!~5@3$W60)PLl zQ7nE_<>l|D7(0rgna~;mXmZqXwc>Qa*|4253zExV;^+FXg(Q`-pSR!lbZ+Zm=kY6G8T06+^?$XL17pA`rvvh7)MCli@tjmB1=b>U&LLyUAO4P> zxU=BkOW#xAhyh(0+l^8(2wIWLG@~^BlwVQ7+FN(13)dGcmVzI*(uiMjJTf$S^hhF{@!0~Wl)(wp(~nEuV~rrUFG#W%9)7Q{jMuY+<)Dd0D%g{^u5#blf31pDY<~2jv(=6Er6&1*u0n|9DL?<2}{q z^{)1^EqmFz>aHygl^f}Cdw=4rz|4&@ZRZs@GWHC~YlDt4J{J_cUKVd4oqXkd)*CkH z@(*v!;EfHu38xuQN_%f?5Y3ZuSRm}_xS$hSrG}y)moEoBn-BW2oz7d(XM2^mkKW24 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= \bigcup\limits_{i;s_i >0} A_i$.\newline - Schrift extrem unleserlich!!! + Sei $\zeta = \sum\limits_{i=1}^{k} s_i \psi_{A_i}$ einfach $\implies \{\zeta > 0\} = \bigcup\limits_{i;s_i >0} A_i$.\newline + Ist $\mu (\{ \zeta >0\}) = \infty$ $\implies s_i \mu (A_i) = \infty$ für mindestens ein $i$ und damit $I(\zeta )= \infty$. \newline + Sei nun $\mu(\{\zeta\}) < \infty$ und $\zeta = \sum\limits_{j=1}^l t_j \psi_{B_j}$ eine zweite einfache Darstellung. \newline + z.z. $\sum\limits_{i=1}^k s_i \mu (A_i) = \sum\limits_{j=1}^l t_j \mu (B_j)$ \newline + $\implies I$ wohldefiniert $\implies A_i\cap B_j$ sind messbar und paarweise disjunkt und es gilt \newline + $0 = \sum\limits_{i=1}^k s_i \psi_{A_i} - \sum\limits_{j=1}^l t_j \psi_{B_j} = \sum\limits_{i=1}^k \sum\limits_{j=1}^l (s_i - t_j )\psi_{A_i \cap B_j}$. \newline + Für $A_i \cap B_j \neq \varnothing \implies s_i = t_j$ \newline $ \implies \sum\limits_{i=1}^{k} s_i \mu (A_i) - \sum\limits_{j=1}^l t_j \mu (B_j) = \sum\limits_{i=1}^k \sum\limits_{j=1}^l (s_i - t_j)\mu (A_i\cap B_j)=0$ \newline + Sei nun $\zeta = \sum\limits_{i=1}^k s_i \psi_{A_i}$, $\Psi = \sum\limits_{j=1}^l t_j \psi_{B_j}$ einfache Darstellungen von $\zeta$,$\psi\in \script{T}^+(\mu) \\ \implies \alpha \zeta + \beta \Psi = \sum\limits_{i=1}^k \sum\limits_{j=1}^l (\alpha s_i + \beta t_j)\psi_{A_i\cap B_j}$. Das ist einfach. \newline + $\implies I(\alpha \zeta + \beta \Psi) = \sum\limits_{i=1}^k \sum\limits_{j=1}^l (\alpha s_i + \beta t_j) \mu (A_i \cap B_j) = \alpha I(\zeta) + \beta I(\Psi) \implies$ i) \newline + In ii) gilt $\Psi - \zeta \in \script{T}^+(\mu)$ und damit $I(\Psi) = I(\zeta + (\Psi - \zeta)) \overset{i)}{=} I(\zeta) + I(\Psi - \zeta) \geq I(\zeta)$ + \end{proof} \begin{remark} @@ -62,7 +71,8 @@ \chapter{Lebesgue-Integral} \end{lemma} \begin{proof} - siehe Aufschrieb + $f$ ist Unterfunktion von $f$ $\implies \int f d\mu \geq I(f)$. \newline + Lemma IV.2 ii) $\implies I(\zeta) \leq I(f)$ $\forall \zeta$ Unterfunktion von f $\implies \int f d\mu \leq I(f)$. \end{proof} \begin{example} @@ -81,16 +91,41 @@ \chapter{Lebesgue-Integral} $\mu = card, X = \mathbb{N}_0$\\ z.z.: $f: \mathbb{N}_0 \to \mathbb{R}$ ist bzgl. $card$ auf $\mathbb{N}_0$ integrierbar $\implies \sum\limits_{k \in \mathbb{N}} f(k)$ absolut konvergent\\ Dann gilt: $\int f d card = \sum\limits_{k \in \mathbb{N}} f(k)$\\ - Beweis siehe Aufschrieb + + i) $f: \mathbb{N}_0 \to [0,\infty]$ \\ + Betrachte $f_n: \mathbb{N}_0 \to \mathbb{R}$, $f_n(k) = \begin{cases} f(k) \text{ ,}k\leq n \\ 0\text{ sonst } \end{cases}$ \\ + $f_n$ sind Unterfunktionen von $f$ mit $I(f_n)=\sum\limits_{k=0}^nf(k) \\ \implies \int f dcard \geq \lim\limits_{n\to\infty} I(f_n) = \sum\limits_{k=0}^{\infty} f(k)$\\ + Umgekehrte Abschätzung: \\ + Dazu sei O.E. $\sum\limits_{k=0}^\infty f(k) <\infty \implies f(k)\to 0$ mit $k\to\infty$ \\ + $\zeta$ Unterfunktion von $f$, so ist $\zeta (k) \neq 0$ nur für endlich viele $k$ und damit $\zeta \leq f_n$ für $n$ hinreichend groß. \\ + $\implies I(\zeta)\leq I(f_n) = \sum\limits_{k=0}^n f(k)\leq \sum\limits_{k=0}^\infty f(k)$ \\ + $\implies \int f dcard \leq \sum\limits_{k=0}^\infty f(k)$ \\ + + Äquivalend von Integrierbarkeit und absolute Konvergenz: \\ + $\inf f^+ dcard + \inf f^- dcard = \sum\limits_{k=0}^\infty f^+(k)+\sum\limits_{k=0}^\infty f^-(k) = \sum\limits_{k=0}^\infty |f(k)|$ \\ + Und weiter \\ + + ii) $f: \mathbb{N}_0 \to \mathbb{\bar{R}}$ \\ + $\int f dcard = \inf f^+ dcard - \int f^- dcard = \sum\limits_{k=0}^\infty f^+(k) - \sum\limits_{k=0}^\infty f^-(k) = \sum\limits_{k=0}^\infty f(k)$ \end{example} - + \newpage \begin{theorem} $f,g:X \to \bar{\mathbb{R}}$ $\mu$-messbar. Ist $f \leq g$ $\mu$-fast überall und $\int f^- d\mu < \infty$, so existieren beide Integrale und es ist: $\int f d\mu \leq \int g d\mu$\\ \glqq$\geq$\grqq gilt entsprechend wenn $f^+ d\mu < \infty$ \end{theorem} \begin{proof} - siehe Aufschrieb + \item[i)] $f,g \geq 0$. Ist $\zeta = \sum\limits_{i=1}^k s_i \psi_{A_i}$ Unterfunktion von $f$ $\implies \Psi := \psi_{\{f \leq g \}}\zeta = \sum\limits_{i=1}^k s_i \psi_{\{f\leq g\}\cap A_i}$ Unterfunktion von g \\ + $\implies I(\zeta) = \sum\limits_{i=1}^k s_i \mu (A_i) = \sum\limits_{i=1}^k s_i \psi(\{f \leq g\}\cap A_i) = I(\Psi) \leq \int g d\mu$ \\ + $\implies\int f d\mu \leq \int g d\mu$ + \item[ii)] $f,g$ beliebig. \\ + Es gilt: $f^+ > 0 \implies f= f^+$ \\ + $f^+ > g^+ \implies f = f^+ > g^+ \geq g \\ + f^- < g^- \implies f \geq f^- > -g^- = g$ \\ + Aus $f \leq g$ folgt $f^+ \leq g^+$ und $f^- \geq g^-$ fast überall. \\ + $\implies \int f^+ d\mu \leq \int g^+ d\mu$, $\infty > \int f^- d\mu \geq \int g^- d\mu \\ + \implies \int f d\mu = \int f^+ d\mu - \int f^- d\mu \leq \int g^+ d\mu - \int g^- d\mu = \int g d\mu$ + \end{proof} \begin{remark} @@ -101,9 +136,20 @@ \chapter{Lebesgue-Integral} \sidenote{Vorlesung 12}{11.11.2020} \begin{remark} - Einschub: zum Beweis von Satz III.7 - siehe Aufschrieb + Einschub: zum Beweis von Satz III.7 \\ + $D$ $\lambda^n$-messbar \\ + Schreibe $D = \bigcup\limits_{j=1}^\infty D_j$ mit $D_j = \{x\in D: j-1 \leq ||x|| < j\}$ \\ + Lemma III.6 $\implies \exists U_{i,j}$ offen bzw. $K_{i,j}$ kompakt mit $K_{i,j} \subset D_j \subset U_{i,j}$ und \\ $\lambda^n (U_{i,j}) < \lambda^n (D_j) + \frac{2^{-j}}{i}$, $\lambda^n (K_{i,j}) > \lambda^n (D_j) - \frac{2^{-j}}{i}$ \\ + $\implies U_i := \bigcup\limits_{j=1}^\infty U_{i,j}$ offen und $A_i := \bigcup\limits_{j=1}^\infty K_{i,j}$ abgeschlossen. \end{remark} + \begin{example} + $K_n = \{\frac{1}{n}\}$, $\bigcup\limits_{n=1}^\infty \{ \frac{1}{n}\}$ nicht abgeschlossen. \\ + \item[Beweis] Sei $x\in \bar{A}_i \implies \exists K\subset \mathbb{R}^n$ kompakt mit $x\in $\r{K} (z.B. $K = \overline{B_{\frac{1}{2}}(x)}$). \\ + $\implies \exists x_n \in A_i$ mit $x_n \to x$ O.E. $x_n \in A_i\cap K$ \\ + $K$ schneidet höchstens endlich viele $K_{i,j}$ \\ + endliche Vereinigung von diesen $K_{i,j}$ und $K$ ist abgeschlossen \\ + $\implies x\in A_i \implies A_i = \overline{A_i} \implies A_i$ abgeschlossen. + \end{example} \begin{lemma}[Tschebyscheff-Ungleichung] Für $f:X \to [0, \infty]$ $\mu$-messbar mit $\int f d\mu < \infty$ gilt: @@ -116,7 +162,7 @@ \chapter{Lebesgue-Integral} \end{lemma} \begin{proof} - siehe Aufschrieb + Für $s \in (0,\infty)$ ist $s \psi_{}\ f \geq s {\}}$ eine Unterfunktion von $f$ \\ $\implies s \mu (\{f = \infty\}) \leq s \mu (\{f\geq s \}) = I(s\psi_{\{f\geq s\}}) \leq \int f d\mu$ \end{proof} \begin{lemma} @@ -128,7 +174,8 @@ \chapter{Lebesgue-Integral} \end{lemma} \begin{proof} - siehe Aufschrieb + \item[i)] Folgt mit $s=\infty$ aus Lemma IV.7 angewand auf $f^+$ + \item[ii)] Lemma IV.7 $\implies \mu(\{f\geq s\})=0$ für $s>0$ $\overset{\text{Lemma II.8}}{\implies} \mu(\{f >0\})=0$ \end{proof} \begin{theorem} @@ -136,7 +183,13 @@ \chapter{Lebesgue-Integral} \end{theorem} \begin{proof} - siehe Aufschrieb + Sei $c_k > 0$ eine Nullfolge mit $\sum\limits_{k=1}^\infty c_k = \infty$ (z.B. $c_k = \frac{1}{k}$) \\ + Setze $f_0 := 0$ und definiere für $k\geq 1$ induktiv $E_k = \{f_{k-1}+c_k \leq f \}$ sowie \\ $f_k = f_{k-1}+c_k\psi_{E_k} \implies f_k = \sum\limits_{j=1}^k c_j\psi_{E_j}$ \\ + $\implies f_k$ sind Treppenfunktionen mit $f_0 \leq f_1 \leq ...$ und $f_k \leq f$ $\forall$ $k\in\mathbb{N}$. Denn ist $x\in E_k$, so gilt $f_k(x) = f_{k-1}(x)+c_k \leq f(x)$ nach Definition. \\ + Ist $x \notin E_k \implies f_k(x) = f_{k-1}(x) \leq f(x)$. \\ + Nach Induktion $\implies \lim\limits_{j\to\infty} f_k(x) \leq f(x)$ $\forall x\in X$. \\ + Sei $\lim\limits_{l\to\infty} < \infty $. Da $\sum\limits_{k=1}^\infty c_k = \infty$ existiert $\infty$-viele $k\in\mathbb{N}$ mit $x\notin E_k$. \\ + $\implies f_{k-1}(x) > f(x)-c_k$ für $\infty$-viele $k$ $\implies \lim\limits_{k\to\infty}f_k(x) \geq f(x)$ \end{proof} \newpage @@ -148,7 +201,7 @@ \chapter{Lebesgue-Integral} \end{theorem} \begin{proof} - siehe Aufschrieb + \end{proof} \begin{theorem} From b49c8b6103fbe6e8c7a5f9bc394eb7ed73b59864 Mon Sep 17 00:00:00 2001 From: Liz Date: Thu, 18 Feb 2021 00:02:56 +0100 Subject: [PATCH 2/6] =?UTF-8?q?S=C3=A4tze=20Kapitel=204?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- analysis_3_skript.pdf | Bin 628406 -> 636128 bytes chapter4.tex | 58 ++++++++++++++++++++++++++++++++++++++---- 2 files changed, 53 insertions(+), 5 deletions(-) diff --git a/analysis_3_skript.pdf b/analysis_3_skript.pdf index f3287bb0156d91fbec58fbd903ddb9c4f836d0cb..29a02a41cc5e904ad1bc0709fa1a668846c59eb2 100644 GIT binary patch delta 59411 zcmXV$Q*@H&W&MgfYz~Cn*iO9{qGal;>K$(v)P}Ch+--O$rwf z-xq>m#{h4G!rr8mK~kdq_qgEMf#b(^mJzAoWaas}m5(0Fuz9JS<{ikbbwsf11<#xo zR#ap$aOayUe~0$EviIYww~L(l`75ZKr?tnkCu86XpMLD(<@Wtm@Z5uy%qfw%*buzp z#+;A8_!(YivsTs^rIb(W@5a&;1rGyv>p4A^>*m}9r(_Qi>M9pf}ONu=DW81~S^S+gA6NdH^cXDbEKEb3YjUWSG!XyeNIsJDa z_a!C9Bn<`rcCFC*P4oxav$Tqn=Uda)w(ky3Q5mgvw71NJFb{(vMDXbaC_V zD5TDh2t5xMxC)ez-!YW!mDX?;UoN5RxTnl(yCoquV!d5MAU{FJ<{nO7jaixxc3*Jj zlhj&l@1;8W>J!Khvy5jyddjxD0u`NIRl94pXN-Urgc%920YfqyX2l z4U$}n6`&2c9vev1cknd0?F!2?Lnn-EN}e~sbQgd-0c-_1Q2ur?A&88&MK6KKaV#-0 zu?UUJIHEQ1Dh0bdomx~c;^*BQABOT4qifIb7_w28=XcNn0ZVBD&3@gY5m3v=K+dpI z>=BAS;SIX$FB02lc1{S(p1f#`Y{mV>;hF;tM!NnCHf&iwgk?db3JIl_AA6L?09Qhj z?gwi^6J2!DZZPK@(JUn~%yTaXIVJ}!=9MhulJ()lo!Nj<&GAZWv`xcqWwGBbjZomC zM7czuAUSA8LEoKT+D3HhR5sw@f;3c#*aMQUY;7$s0R{k%+Cf9Zm!O(JP0dyEF`WX9 zr&fx@fyJAP)^H=?qB6!(ComeI{31gVMS}Q8jU_O z_mHhlv?V-u&RK7)Zi-hf$2Y}+MS}X#^n-?Zzk-JLy%p)CSCaWz=Ih$6Qm&!&&6@&5 z;}RXV?2i+XV1me0A<`h^Sn!xoF{*+sjT3sWKfc|*8IdwS7na>wW~-S3Zn-NY6hxO^ zZe^tkFUd4gm*s3K!PAf!Vpd^Z`nXSdxqzA#qO#`nT3R07k0zB}veu

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z=nahvfG=wb@j)byCKX^*%`-`Tz6x-Hp-V4%VN(lm@$pdbXo71U+-L<_E~GG6_O&~^ zuEkCZGvHb|>YYaJZk3H3B_WNYP9I8N*a?bcIul)wL3 zE=UoV&x@42ag$)Gh91io 0$ ist $\lim\limits_{k\to\infty} f_k(x) = f(x)$ $\forall x\in E_i$. \\ + Außerdem: $E_{i,1} \subset E_{i,2} \subset ..." \overset{\text{Satz I.7}}{\implies} \mu (E_i) = \lim\limits_{k\to\infty}\mu (E_{i,k})\\ + \implies \Theta I(\zeta) = \sum\limits_{i=1}^m \Theta s_i \mu (E_i) = \lim\limits_{k\to\infty}^m \Theta s_i \mu (E_{i,k}) \leq \lim\limits_{k\to\infty} \int f_k d\mu$ \\ + Mit $\Theta \to 1$ und nach Bildung des Supremums über alle Unterfunktionen $\zeta$ gilt: \\ $\int f d\mu \leq \lim\limits_{k\to\infty} \int f_k d\mu$. \end{proof} \begin{theorem} @@ -212,7 +221,35 @@ \chapter{Lebesgue-Integral} \end{theorem} \begin{proof} - siehe Aufschrieb + \item[I.] $f\geq 0$, $\alpha > 0$ $\implies \int (\alpha f) d\mu = \alpha \int f d\mu$. \\ + Sei dazu $\zeta$ Unterfunktion von $f$ $\implies \alpha \zeta$ ist Unterfunktion von $\alpha f$ und mit Lemma IV.2 folgt: $\alpha I(\zeta) = I(\alpha \zeta) \leq \int (\alpha f) d\mu$. Also: $\alpha \int f d\mu \leq \int (\alpha f) d\mu$. \\ + Ersetze $\alpha$ durch $\frac{1}{\alpha}$ von $f$ durch $(\alpha f) \implies \alpha \int f d\mu \geq \int (\alpha f) d\mu$. + \item[II.] $f$ integrierbar, $\alpha > 0$ $\implies \alpha \int f d\mu = \int (\alpha f) d\mu$. \\ Denn $(\alpha f)^\pm = \alpha f^\pm \\ \implies \int (\alpha f) d\mu = \int (\alpha f)^+ d\mu - \int (\alpha f)^- d\mu \overset{\text{I}}{=} \alpha \int f^+ d\mu - \alpha \int f^- d\mu = \alpha \int f d\mu$. + \item[III.] $\int (-f) d\mu = -\int f d\mu$, denn $(-f)^\pm = f^\mp \\ \implies \int (-f) d\mu = \int (-f)^+ d\mu - \int (-f)^- d\mu = \int f^- d\mu - \int f^+ d\mu = -\int f d\mu$ \\ + \item[z.z.] Linearität für $f+g$ + \item[i)]$f,g \geq 0$ \\ + Satz IV.9 $\implies \exists \zeta_k,\Psi_k \in \script{T}^+(\mu)$ mit $\zeta_k \to f$, $\Psi_k \to g$ punktweise.$\implies \zeta_k + \Psi_k \to f+g$ \\ + Satz IV.10 und Lemma IV.2 \\ + $\int (f+g) d\mu = \lim\limits_{k\to\infty} \int (\zeta_k +\Psi_k ) d\mu = \lim\limits_{k\to\infty}(\int \zeta_k d\mu + \int \Psi_k d\mu) = \int f d\mu + \int g d\mu$ + \item[ii)] $f,g: X\to\mathbb{R}$ integrierbar \\ + Betrachte $\phi = f^+ + g^+ - (f+g)^+ \geq 0$, $\psi = f^- + g^- -(f+g)^- \geq 0$ \\ + $\implies \phi-\psi = f^+ - f^- +g^+-g^--(f+g)^++(f+g)^- = f+g-(f-g) = 0 \\ + \implies \int (f^+ + g^+) d\mu \overset{\text{i)}}{=} \int (f+g)^+ d\mu + \int \phi d\mu $ \\ + $\implies \int (f^- + g^- )d\mu \overset{\text{i)}}{=} \int (f+g)^- d\mu + \int \psi d\mu$ \\ + $\implies$ Beh. + \item[iii)] $f,g: X\to\mathbb{\bar{R}}$ \\ + O.E. $\int f d\mu + \int g d\mu > -\infty.$ \\ + Also $\int f^- d\mu + \int g^- d\mu < \infty$. \\ + (Sonst gehe zu $-f$, $-g$ über und benutze den ersten Teil mit $\alpha = -1$) \\ + Lemma IV.8 $\implies \mu(\{f=-\infty \} \cup \{g = -\infty \}) =0\implies \{(f,g) = \pm (\infty,-\infty) \}\in \script{M}(\mu)$ ist eines $\mu$-Nullmenge. \\ + Aus $(f+g)^- \leq f^- + g^-$ und Monotonie folgt: \\ + $\int (f+g)^- d\mu \leq \int (f^- + g^-) d\mu \overset{\text{i)}}{=} \int f^- d\mu + \int g^- d\mu < \infty \implies \int (f+g) d\mu$ ist definiert. \\ + Es gilt: \\ + $(f+g)^+ - (f+g)^- = f+g = f^+ + g^+ -(f^- + g^- )$ bzw. \\ + $(f+g)^+ + f^- +g^- = (f+g)^- +f^+ + g^+ \\ + \overset{\text{i)}}{\implies} \int (f+g)^+ d\mu + \int f^- d\mu + \int g^- d\mu = \int (f+g)^- d\mu + \int f^+ d\mu + \int g^+ d\mu$.\\ + Ist $\int f^+ d\mu + \int g^+ d\mu < \infty$, so sind alle Integrale endlich und \\ $\int (f+g) d\mu = \int (f+g)^+ d\mu - \int (f+g)^- d\mu = \int f^+ d\mu + \int g^+ d\mu - \int f^- d\mu - \int g^- d\mu = \int f d\mu + \int g d\mu$. \\ + Ist $\int f^+ d\mu + \int g^+ d\mu = \infty \implies \int (f+g)^+ d\mu = \infty \\ \implies \int (f+g) d\mu = \infty = \int f d\mu + \int g d\mu$ \end{proof} \begin{definition} @@ -234,10 +271,19 @@ \chapter{Lebesgue-Integral} \int\limits_{\mathbb{R}^n \setminus B_1(0)} f d\lambda^n < \infty &\Leftrightarrow \alpha > n\\ \int\limits_{B_1(0)} f d\lambda^n < \infty &\Leftrightarrow \alpha < n \end{align*} - Beweis siehe Aufschrieb + Vergleiche dazu $f$ mit $g = \sum\limits_{k\in\mathbb{Z}} 2^{-k \alpha} \psi_{A_k}$, $A_k = \{2^k \leq ||\alpha|| < 2^{k+1} \}$ \\ + Es gilt: \\ + $2^{-\alpha} g \leq f \leq g$ für $\alpha > 0$ \\ + $2^{-\alpha} g \geq f \geq g$ für $\alpha \leq 0$ \\ + Monotonie $\implies$ Reicht die Aussagen für $g$ zu zeigen. \\ + $A_k = 2^k A_0, Y_n := \lambda^n (A_0) \in (0,\infty) \\ + \overset{\text{Satz III.16}}{\implies} \lambda^n (A_k) = (2^l) \lambda^n(A_0) = 2^{nk}Y_n\\ + \sum\limits_{k=0}^l 2^{-k\alpha}\psi_{A_k}$ konv. punktweise auf $\mathbb{R}^n$ gegen $g\psi_{\mathbb{R}^n\setminus B_1(0)}$. Folge ist monoton wachsend. \\ + $\overset{\text{Satz über Mon. Konvergenz}}{\implies} \int\limits_{\mathbb{R}^n\setminus B_1(0)} g d\lambda^n = \sum\limits_{k=0}^\infty \int 2^{-k\alpha}\psi_{A_k} d\lambda^n = Y_n \sum\limits_{k=0}^\infty 2^{n-\alpha)k} = \begin{cases}Y_n \frac{1}{1-2^{n-\alpha}} \text{falls } \alpha \geq n \\ \infty \text{ sonst } \end{cases}$ \\ + Entsprechend gilt auf $B_1(0)$ \\ + $\int\limits_{B_1(0)}g d\lambda^n = \sum\limits_{-\infty}^\infty \int 2^{-k\alpha}\psi_{A_k} d\lambda^n = Y_n \sum\limits_{k=-1}^\infty 2^{(n-\alpha)k} = \begin{cases} Y_n \frac{1}{2^{n-\alpha}-1} \text{ falls } \alpha < n \\ \infty \text{ sonst }\end{cases}$ \end{example} - \newpage \sidenote{Vorlesung 13}{14.12.20} \begin{theorem} Sei $f: X \to \bar{\mathbb{R}}$ $\mu$-messbar. Dann gelten: @@ -249,7 +295,9 @@ \chapter{Lebesgue-Integral} \end{theorem} \begin{proof} - siehe Aufschrieb + Es gilt $|f| = f^+ + f^-$ und Satz IV.11 impliziert $\int |f| d\mu = \int (f^+ + f^-) d\mu = \int f^+ d\mu + \int f^- d\mu \implies \text{i)} $ \\ + Ist $\int f d\mu$ definiert, so gilt $| \int f d\mu | = |\int f^+ d\mu - \int f^- d\mu | \leq \int f^+ d\mu + \int f^- d\mu = \int |f| d\mu \implies \text{ii)}$ \\ + Sei $g$ wie in iii) $\overset{\text{Satz IV.6}}{\implies} \int |f| d\mu \leq \int g d\mu \implies f$ ist integrierbar. \end{proof} \begin{example} From bb0f7d5f99f1c377664a35ccc34c6bc275ac71d3 Mon Sep 17 00:00:00 2001 From: Liz Date: Thu, 18 Feb 2021 12:08:40 +0100 Subject: [PATCH 3/6] Bis inklusive VL 14 --- analysis_3_skript.pdf | Bin 636128 -> 647726 bytes chapter5.tex | 70 ++++++++++++++++++++++++++++++++---------- 2 files changed, 53 insertions(+), 17 deletions(-) diff --git a/analysis_3_skript.pdf b/analysis_3_skript.pdf index 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z%0|(!p#fpl&(18X3g`^x;6!z;Fyo@U%GWG(%MRvDGh(n69hQfEgnUf?_YOw?!T9D9fff%3i!iGoizq9LfDj7@E2oek zGdl;PfCv*i6DzAICp!}#!T+D~Z$4>b8&fAU0?vOk5^O_J1Ej1}RDK8jIODx2cy4Ig zPBc9l3luhW=Bd&&Em~4~%fA;MH8(a=#RdMMIu)14{LO-l%qombDhsQC!h%KKBM=of zY(@k{lAanG1`<#PkMk?waBBtT8ls*3~=44wD%?Ha7b6dvctX;V~qW zqnAn`5qY>S008MRJ*GGe|IYM_X41kX8RhTziF#2e3oRp3pX9e{jZwUKZ@Fv(JFzbG zvT`hGerFJcU)2VoR4_$Q6Gr0(toIQP%DCPe|4~O$sW03Qi95CeRK{4TwV$x{ro%)1-X1IWx`}yAG{XN^)DM@z<2D#I{r;h_#&5wwM(}yKd5^fD=%c zoVmOACEz1WR_{!%`l#duXr8c0g2UK%N( zSGOu2%z)qDDB=*7F^b?L;*iE3sIQ!EtI2&l05FK-50_&WV=)e9fG045a!<(|tamG- z0bDqvyAbIAn;;b{(a4TR@b(rVJ~H;77Q@%P>3M7z#VD37XjiW#b;rj|pO->#X~7nV z(la(PaZ?2OK`bFUI)S*I8h1$g2puR@gGuzyF4|?7zWe4u+P@xhmW^J;!lhXLb^vEjNvm3m4ma4 0$. \\ + Sind $Z_1$, $Z_2$ bel. Zerlegungen $\implies \bar{f}_{Z_2}(x) - \underline{f}_{Z_1}(x) \geq s$ $\forall x\in N_f(s)$. \\ + $\implies \bar{S}_{Z_2}(f) - \underline{S}_{Z_1}(f) = \int\limits_{I} (\bar{f}_{Z_2}(x)-\underline{f}_{Z_1}(x)) d\lambda^1 \geq s \lambda^1 (N_f(s))$ \\ + Ist $f$ Riemann-integrierbar, so bilde Infimum über alle $Z_1$, $Z_2$ und schließe $\lambda^1 (N_f(s))=0$ $\forall s>0$ $\implies$ "$\implies$". \\ + Sei nun $f$ $\lambda^1$-fast überall stetig und $Z_i$ Folge von Zerlegungen mit Feinheit \\ $\delta_i := \max\limits_{1\leq j \leq N_i} |x_{i,j}-x_{i,j-1}| \to 0$ \\ + Ist $f$ stetig in $x$, so folgt $\bar{f}_{Z_i} (x) \geq \inf\limits_{|y-x| \leq \delta_i } f(y) \to f(x)$ und $\underline{f}_{Z_i} (x) \geq \inf\limits_{|y-x| \leq \delta_i } f(y) \to f(x)$ mit $i\to\infty$ $\implies \bar{f}_{Z_i}$, $\underline{f}_{Z_i}$ konvergiert punktweise $\lambda^1$-fast überall gegen $f$ $\implies f$ ist $\lambda^1$-messbar nach Kapitel I. \\ + Aus $|\bar{f}_{Z_i}|$, $|\underline{f}_{Z_i}| \leq \sup\limits_{I} |f| < \infty$ folgt aus Satz V.2 $\bar{S}_{Z_i}(f) = \int\limits_{I} \bar{f}_{Z_i} d\lambda^1 \to \int\limits_{I} f d\lambda^1$, \\ + $\underline{S}_{Z_i}(f) = \int\limits_{I} \underline{f}_{Z_i} d\lambda^1 \to \int\limits_{I} f d\lambda^1 \implies f$ ist Riemannint. mit $\int\limits_{a}^b f(x) dx = \int\limits_{I} f d\lambda^1$. \end{proof} \begin{theorem} @@ -60,7 +70,9 @@ \chapter{Konvergenzsätze und $L^n$-Räume} \end{theorem} \begin{proof} - siehe Aufschrieb + Sei $\alpha_k \to x_0$ Folge $\implies \exists \mu$-Nullmenge $N$, so dass $\forall y\in y\setminus N$ gilt: \\ + $f(x_k,y) \to f(x_0,y)$ und $|f(x_k,y)| \leq g(y)$ $\forall k\in\mathbb{N}$. \\ + $\overset{\text{Satz V.2}}{\implies} F(x_0) = \int f(x_0,y) d\mu(y) = \lim\limits_{k\to\infty} \int f(x_k,y) d\mu(y) = \lim\limits_{k\to\infty} F(x_k)$. \end{proof} \sidenote{Vorlesung 14}{18.12.20} @@ -78,7 +90,10 @@ \chapter{Konvergenzsätze und $L^n$-Räume} \end{align*} \end{theorem} \begin{proof} - siehe Aufschrieb + Zu jeder Folge $x_k \to x_0$ existiert $\mu$-Nullmenge $N\subset Y$, sodass $\forall y\in Y\setminus N$ gilt: \\ + $$\lim\limits_{k\to\infty} \frac{f(x_k,y)-f(x_0,y)}{x_k-x_0} = \frac{\partial f}{\partial_x}(x_0,y)$$ \\ + $$\frac{|f(x_k,y)-f(x_0,y)|}{|x_k-x_0|} \leq g(y)$$ + $$\overset{\text{Satz V.2}}{\implies} \frac{F(x_k)-F(x_0)}{x_k-x_0} = \int \frac{f(x_k,y)-f(x_0,y)}{x_k-x_0} d\mu(y) \to \int \frac{\partial f}{\partial x}(x_o,y) d\mu(y)$$ \end{proof} \newpage @@ -96,7 +111,10 @@ \chapter{Konvergenzsätze und $L^n$-Räume} \end{lemma} \begin{proof} - siehe Aufschrieb + Nach Vor. gilt $\forall y\in Y$ mit Ausnahme einer $\mu$-Nullmenge $N$: \\ + $$ \frac{|f(x+hc_i, y)-f(x,y)|}{h} \leq \int\limits_{0}^1 \left| \frac{\partial f}{\partial x_i} (x+tc_i,y) \right| dt \leq g(y) $$ + Satz V.5 $\implies$ $F$ ist in allen $x\in \script{U}$ nach $x_i$ partiell differenzierbar mit gewünschten Ableitung. \\ + Satz V.4 $\implies$ partielle Ableitungen sind stetig auf U $\implies F\in C^1(\script{U})$. \end{proof} \begin{example} @@ -113,13 +131,16 @@ \chapter{Konvergenzsätze und $L^n$-Räume} &= -1 - t^2 F'(t) \end{align*} $\implies F'(t) = \dfrac{-1}{1+t^2}$\\ - \newline - ... (siehe Aufschrieb)\\ - \newline - $\int\limits_0^{\infty} \dfrac{\sin(x)}{x} dx = \dfrac{\pi}{2}$ + Weiter ist $\lim\limits_{t\to\infty} f(t,x) = 0$ $\forall x>0$ mit Majorante $\text{e}^{-x}$ \\ + Satz V.2 $\implies \lim\limits_{t\to\infty} F(t) = 0 \implies F(t) = \frac{\pi}{2}-\arctan{t}$ $\forall t>0$. \\ + Für $t>0$ und $0\infty \implies ||f||_{L^p}$ ist wohldefiniert nach Satz IV.6. \\ + 1. folgt aus Lemma IV.8 \\ + 2. folgt aus Linearität des Integralt. \\ + $t \mapsto t^p$ ist konvex auf $[0,\infty) \implies |f+g|^p = 2^p \left| \frac{f+g}{2}\right| ^p \leq 2^{p-1} (|f|^p + |g|^p) \implies$ Aus $f,g \in L^p(\mu)$ folgt $f+g\in L^p(\mu)$. \\ + $\Delta$-Ungleichung folgt später. \\ + \item[$p=\infty$:] + \item[1)] ist klar + \item[2)] O.E. $\lambda >0 \rightarrow \{|\lambda f| > \lambda s\} = \{|f| > s \}$ + \item[3)] $\{|f+g| > s_1+s_2 \}\subset \{|f|>s_1 \}\cup \{|g|>s_2 \}$ \end{proof} \begin{lemma}[Youngsche Ungleichung] @@ -158,17 +187,19 @@ \chapter{Konvergenzsätze und $L^n$-Räume} \end{lemma} \begin{proof} - siehe Aufschrieb + Sei $y\geq 0$ fest und $f(x) = \frac{1}{p}x^p +\frac{1}{q}y^q -xy \implies f'(x) = \alpha^{p-1}-y \begin{cases} <0 \text{ für } x 0 \text{ für } x>y^{\frac{1}{p-1}}\end{cases} $ \\ + $\implies \forall x\geq 0$: $f(x) \geq f(y^{\frac{1}{p-1}}) = \frac{1}{p} y^{\frac{p}{p-1}}+\frac{1}{q} y^{\frac{p}{p-1}} - y^{\frac{p}{p-1}} = 0$ \end{proof} - \newpage \begin{theorem}[Höldersche Ungleichung] Für $\mu$-messbare $f,g: X \to \bar{\mathbb{R}}$ gilt: \ \ $|\int fg d\mu| \leq ||f||_{L^p} ||g||_{L^p}$,\\ falls $1 \leq p,q \leq \infty$ mit $\dfrac{1}{p} + \dfrac{1}{q} = 1$ \end{theorem} \begin{proof} - siehe Aufschrieb + O.E. $f,g \geq 0$ und $||f||_{L^p} = ||g||_{L^q} = 1$ (sonst $\tilde{f} = \dfrac{f}{||f||_{L^p}}$, $\tilde{g} = \dfrac{g}{||g||_{L^q}}$) \\ + Lemma V.10 $\implies \int fg d\mu \leq \int \left(\frac{f^p}{p}+\frac{g^q}{q} \right)d\mu = 1 = ||f||_{L^p} ||g||_{L^q} \\ \implies$ Beh. für $1 < p,q < \infty$. \\ + Fall $p=1, q =\infty$ folgt sofort aus Satz IV.6. \end{proof} \begin{theorem}[Minkowski-Ungleichung] @@ -176,7 +207,7 @@ \chapter{Konvergenzsätze und $L^n$-Räume} \end{theorem} \begin{proof} - siehe Aufschrieb + $$ ||f+g||^p_{L^p}$$ $$= \int |f+g|^p d\mu \leq |f| |f+g|^{p-1} d\mu + \int |g| |f+g|^{p-1}d\mu$$ $$\leq ||f||_{L^p}||f+g||_{L^p}^{p-1} + ||g||_{L^p}||f+g||_{L^p}^{p-1}$$ $$ \overset{\text{Kürzen}}{\implies} ||f+g||_{L^p} \leq ||f||_{L^p} + ||g||_{L^p}$$ \end{proof} \begin{lemma} @@ -189,7 +220,12 @@ \chapter{Konvergenzsätze und $L^n$-Räume} \end{lemma} \begin{proof} - siehe Aufschrieb + Betrachte $g_k := \sum\limits_{j=1}^k |u_j|$, $g:= \sum\limits_{j=1}^\infty |u_j|$ \\ + Es gilt: $g_1 \leq g_2 \leq ...$ und $g_k(x) \to g(x) \in [0,\infty]$ mit $k\to\infty$ $\forall x\in X$ \\ + Satz IV.10 $\implies ||g||_{L^p} = \lim\limits_{k\to\infty} ||g_k||_{L^p} \overset{\text{Satz V.12}}{\leq} \sum\limits_{j=1}^\infty ||u_j||_{L^p} < \infty$ \\ + Lemma IV.8 $\implies N:=\{g=\infty \}$ ist $\mu$-Nullmenge. \\ Für $x\in X\setminus N$ ist $\sum\limits_{j=1}^\infty u_j(x)$ absolut konvergent.$\implies (f_k(x))$ ist Cauchy-Folge in $\mathbb{R} \\ + \implies f(x) = \lim\limits_{k\to\infty} f_k(x)$ existiert $\forall x\in X\setminus N$. \\ Weiter ist $|f_k|^p \leq |g|^p \in L^1(\mu)$, sowie $|f-f_k|^p \leq 2^{p-1}(|f|^p+|f_k|^p) \leq 2^p g^p$ \\ + Satz V.2 $\implies f\in L^p(\mu)$ und $||f-f_k||_{L^p} \to 0$ \end{proof} \sidenote{Vorlesung 15}{21.12.20} From c5f09a617d8445ff01b1b3857e49de189b88ac03 Mon Sep 17 00:00:00 2001 From: Liz Date: Sat, 20 Feb 2021 14:17:24 +0100 Subject: [PATCH 4/6] Kap 5 --- analysis_3_skript.pdf | Bin 647726 -> 664571 bytes chapter5.tex | 92 ++++++++++++++++++++++++++++++++++++++---- chapter6.tex | 29 ++++++++++++- 3 files changed, 111 insertions(+), 10 deletions(-) diff --git a/analysis_3_skript.pdf b/analysis_3_skript.pdf index 517457dd16f4f06b59719f582af54676dac1d01a..dfcdfcf511c91ab9fadd2b3b98cd993cb95af9b8 100644 GIT binary patch delta 86648 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z)bQ^W#cPo^+*@0F)!R!8#`A&H4-$jPqpeATU} z&ajlS8C?>kLQP)E#ew=m;3rN4Dz{KnfkJe_4B!>694hjzsT~W8gZQHG{4XH z$Byy!t5Li7Z=CHv=$tzJ=&+#{5p4HbsrzQ8Du#fp+4$c?K69GE7N()pS$%QQy25|V z9R&IiZKAQ5fO#s5#c(>Bk9$l*v6}+7DO(F}s|DbE=Y6lo0)Z$RrV*eVDz-#ov(ii~ zE;ehqzKB}klTH;!+DON~CuE{d>72RDU+RYLs9RA#Td+bYL~HrSq2lz6>Meuml7ik^ zay`*m8iZSMsV^~y{R61|Chr=XQX~lw-F)#XqVSS{sneG#C zqn2`Mb-U~R?&5~?%bO4IO#ipx*~Rz2PyVh@P!P-|5EdHf;R%v~$Vf{_g82EhO?5#3 E1%@I15C8xG diff --git a/chapter5.tex b/chapter5.tex index bdc77dd..696153a 100644 --- a/chapter5.tex +++ b/chapter5.tex @@ -233,8 +233,22 @@ \chapter{Konvergenzsätze und $L^n$-Räume} \begin{theorem}[Satz von Riesz-Fischer] $(L^p(\mu), ||\cdot||_{L^p})$ ist vollständig, also ein Banachraum. $(1 \leq p \leq \infty)$ \end{theorem} + \begin{proof} - siehe Aufschrieb + Sei $f_k \in L^p(\mu)$ Cauchyfolge bzgl $||\cdot ||_{L^p}$. \\ + Es reicht zu zeigen: $\exists$ Teilfolge, welche in $L^p(\mu)$ konvergiert. + \item[1)] $1 \leq p < \infty$: \\ + Nach Wahl einer Teilfolge sei $||f_{k+1}-f_k||_{L^p} \leq 2^{-k}$ $\forall k\in\mathbb{N}$. \\ + Mit $f_0 := 0 \implies f_k = \sum\limits_{j=1}^k u_k$ mit $u_j = f_j - f_{j-1}$. \\ + Lemma V.13 $\implies$ $f_k$ konvergiert in $L^p(\mu)$ bzw. punktweise $\mu$-fast überall gegen $f\in L^p(\mu) \implies$ Beh. + \item[2)] $p = \infty$ \\ + Wegen $\left| ||f_k||_{L^\infty} - ||f_l||_{L^\infty} \right| \leq ||f_k - f_k||_{L^\infty}$ existiert $\lim\limits_{k\to\infty} ||f_k||_{L^\infty}$. \\ + Die Mengen $N_k = \{|f_k| > ||f_k||_{L^\infty} \}$ sowie $N_{k,l} = \{|f_k-f_l| > ||f_k-f_l||_{L^\infty} \}$ haben $\mu$-Maß Null $\implies N = \bigcup\limits_{k=1}^\infty N_k \cup \bigcup\limits_{k,l=1}^\infty N_{k,l}$ ist $\mu$-Nullmenge. \\ + Für $x\in X\setminus N$ gilt: $|f_k(x)-f_l(x)|\leq ||f_k-f_l||_{L^\infty} < \epsilon$ für $k,l \geq k(\epsilon) \\ + \implies f(x) = \lim\limits_{k\to\infty} f_k(x)$ ist definiert $\forall x\in X\setminus N$. \\ + Weiter gilt für $x\in X\setminus N$:\\ + $|f(x)| = \lim\limits_{k\to\infty}|f_k(x)| \leq \lim\limits_{k\to\infty} ||f_k||_{L^\infty}$ und $|f_k(x) - f(x)| = \lim\limits_{l\to\infty} |f_k(x)-f_l(x)| \leq \lim\limits_{l\to\infty} ||f_k-f_l||_{L^\infty} \leq \epsilon$ für $k\geq k(\epsilon)$ \\ + $\implies ||f||_{L^\infty} \leq \lim\limits_{k\to\infty} ||f_k||_{L^\infty} < \epsilon$ und $||f_k-f||_{L^\infty} \to 0$ mit $k\to\infty$. \end{proof} \begin{lemma} @@ -277,20 +291,60 @@ \chapter{Konvergenzsätze und $L^n$-Räume} \begin{theorem} Sei $\Omega \subseteq \mathbb{R}^n$ offen und $1 \leq p < \infty$. Dann existiert zu jedem $f \in C^p(\Omega)$ eine Folge $f_k \in C_c^0(\Omega)$ mit $|| f_k - f||_{L^p(\Omega)} \to 0$ mit $k \to \infty$. \end{theorem} + \begin{proof} - siehe Aufschrieb + \item[1)] $f=\psi_E$ mit $E\subset \Omega$ messbar mit $\lambda^n(E) < \infty$\\ + Sei $\epsilon >0$. Lemma III.6 $\implies$ $\exists K\subset E$ kompakt mit $\lambda^n(E\setminus K) < \frac{\epsilon}{2}$ \\ + Setze $f_\rho: \Omega \to [0,1]$, $f_\rho (x) = \left( 1-\frac{dist(x,K)}{\rho}\right) ^+ \\ + \overset{\text{1),2)}}{\implies} f_\rho \in C^0(\Omega)$, $spt f_\rho = \{x: dist(x,K)\leq \rho \}$ ist kompakte Teilmenge von $\Omega$ für $\rho$ hinreichend klein. $\implies f_\rho \in C^0_c (\Omega)$ für $\rho << 1$ und $f_\rho = f$ auf $K$. \\ + $$\implies \int\limits_{\Omega} |f_\rho -f |^p d\lambda^n \leq 2^{p-1} \int\limits_{\Omega\setminus K} (|f_\rho)|^p + |f|^p d\lambda^n$$ + $$ \leq 2^{p-1} (\lambda^n (\{ 0< dist(\cdot, K)\leq \rho\}) + \lambda^n(E\setminus K))$$ + $$ < 2^{p-1}\epsilon \text{ für } \rho \text{ hinreichend klein }$$ + $\implies$ Beh. + \item[2)] $f$ beliebig, $f\in L^p(\Omega)$ \\ + O.E. $f \geq 0$ sonst betrachte $f^+$ und $f^-$. \\ + Nach Satz IV.9 existiert Folge von Treppenfunktionen $f_1 \leq f_2 \leq ...$ mit \\ $\lim\limits_{k\to\infty} f_k(x) = f(x)$ $\forall x\in \Omega$. \\ + Satz V.2 $\implies f_k \to f$ in $L^p(\Omega)$. \\ + Verwende dazu Majorante $|f-f_k|^p \leq 2^{p-1}(|f|^p+|f_k|^p) \leq 2^p |f|^p \in L^p(\Omega)$. \\ + Für die Treppenfunktion $f_k$ gilt nach Lemma IV.7 + $$ \lambda^n (\{f_k \geq s \}) \leq \frac{1}{s^p} \int\limits_\Omega |f_k|^p d\lambda^n \leq \frac{1}{s^p} \int\limits_\Omega |f|^p d\lambda^n < \infty$$ + Da $f_k$ endliche Linearkombination von charakteristischen Funktionen ist, folgt die Behauptung. \end{proof} \begin{remark} - $BC^0(\Omega)$ bezeichnet die Menge der beschränkten, stetigen Funktionen auf $\Omega$. Mit Supremumsnorm $||\cdot||_{sup}$ ist diese ein Banachraum.\\ - ... (Rest siehe Aufschrieb) + $BC^0(\Omega)$ bezeichnet die Menge der beschränkten, stetigen Funktionen auf $\Omega$. Mit Supremumsnorm $||\cdot||_{\sup}$ ist diese ein Banachraum. Für $f\in BC^0(\Omega)$ gilt \\ + $$\{ |f|>s \} \neq \varnothing \Leftrightarrow \lambda^n(\{ |f|>s \}) > 0$$ + $\implies ||f||_{L^\infty} = ||f||_{\sup}$ $\forall f\in BC^0(\Omega)$. \\ + Angenommen $f\in L^\infty(\Omega)$ kann durch $f_k \in BC^0(\Omega)$ bzgl der $L^\infty$-Norm approximiert werden, dann wäre $f_k$ eine Cauchyfolge bzgl $\sup$-Norm \\ + $\implies \exists \tilde{f} \in BC^0(\Omega)$ mit $f_k\to \tilde{f}=f$ fast überall, aber $L^\infty$-Funktion ohne stetigen Repräsentanten. + \item[] \underline{Fourier-Reihen:} \\ + Betrachte Funktionen $f\in L^2(I,\mathbb{C})$ mit $I =(-\pi, \pi) \Leftrightarrow \text{Re}(f)$, $\text{Im}(f) \in L^2(I)$ \\ + Satz V.14 $\implies L^2(I,\mathbb{C})$ ist vollständig bzgl $L^2$-Norm. Also ein Hilbertraum mit dem hermitischen Skalarprodukt $$_{L^2} = \int\limits_{-\pi}^\pi f(x) \overline{g(x)}dx$$ + $w_k (x) = \frac{1}{\sqrt{2\pi}} \text{e}^{ikx}$, $k\in\mathbb{Z}$ bilden ein Orthonormalsystem und spannen den Raum $\mathbb{P}$ der Trigonometrischen Polynome auf. \\ + Das $n$-te Fourierpolynom $f_n$ ist definiert als die Orthonormalprojektion von $f$ auf den Raum $\mathbb{P}^n$ der trigonometrischen Polynome von Grad $\leq n$. , Also\\ + $$ f_n = \sum\limits_{k=-n}^n _{L^2} w_k = \sum\limits_{k=-n}^n \hat{f}(k) \text{e}^{ik}$$ + $\hat{f}(k) = \frac{1}{\sqrt{2\pi}} _{L^2} = \frac{1}{2\pi} \int\limits_{-\pi}^\pi f(x) \text{e}^{-ikx} dx$. Die Folge $f_n$ heißt Fourier-Reihe von $f$. \\ + Wegen $f-f_n \perp_{L^2} \mathbb{P}_n$ gilt $\forall p\in \mathbb{P}_n$ : \\ + \begin{align*} + (\ast) & & |f-p||_{L^2}^2 = ||(f-f_n)+(f_n-p) ||_{L^2}^2 = ||f-f_n||_{L^2}^2 + ||f_n-p||_{L^2}^2 + \end{align*} + Mit $p = 0$ folgt die Besselsche Ungleichung + \begin{align*} + (\ast\ast) & & 2\pi \sum\limits_{k=-n}^n |\hat{f}(k)|^2 = ||f_n||^2_{L^2} \overset{(\ast)}{\leq} ||f||_{L^2}^2 & & \forall n\in\mathbb{N}_0 + \end{align*} + $\implies f_n$ ist eine $L^2$-Cauchyfolge, denn für $m\geq n$ folgt \\ $||f_m-f_n||^2_{L^2} = 2\pi \sum\limits_{k=n+1}^m |\hat{f}(k) |^2 + 2\pi \sum\limits_{k=-m}^{-(n+1)} |\hat{f}(k)|^2 < \epsilon$ für $n$ groß genug. \\ + $\overset{\text{Riesz-Fischer (Satz V.14)}}{\implies} f_n$ konvergiert in $L^2(I,\mathbb{C})$ gegen eine Funktion $\in L^2(I, \mathbb{C})$. \\ + Frage: Konvergiert $f_n$ gegen $f$? + \end{remark} \begin{theorem} Für $f \in L^2(I, \mathbb{C})$ konvergiert $f_n$ gegen $f$ in $L^2(I, \mathbb{C})$? (bezieht sich auf Bem. vorher) \end{theorem} \begin{proof} - siehe Aufschrieb + Wegen $(\ast)$ gilt $$ ||f-f_n||_{L^2} = \min\limits_{p\in\mathbb{P}_n} ||f-p||_{L^2}$$ + \item[]\underline{z.z.} $\mathbb{P}$ liegt dicht in $L^2(I,\mathbb{C})$ \\ + Satz V.17 $\implies C_c^0(I,\mathbb{C})$ dicht in $L^2(I,\mathbb{C})$ und jede stetige Funktion kann gleichmäßig und damit auch in $L^2$ durch stückweise konstante Funktionen approximiert werden. Für $f$ stückweise $C^1$ haben wir in Analysis II gezeigt: $f_n \to f$ gleichmäßig, also auch in $L^2$. \end{proof} \begin{remark} @@ -298,12 +352,14 @@ \chapter{Konvergenzsätze und $L^n$-Räume} $\ell^2(\mathbb{C})$ ist vollständig (folgt aus Riesz-Fischer angewandt auf das Zählmaß auf $\mathbb{Z}$) \end{remark} - \newpage \begin{lemma} Die Abbildung $\script{F}: (L^2(I, \mathbb{C}), ||\cdot||_{L^2}) \to (\ell^2(\mathbb{C}), ||\cdot||_{\ell^2}), \script{F}(f) = (\hat{f}(k))_{k \in \mathbb{Z}}$ ist eine Isometrie von Hilberträumen. \end{lemma} + \begin{proof} - siehe Aufschrieb + Aus Satz V.18 folgt mit $p=0$ in $(\ast)$: $$ ||\script{F}(f)||_{l^2}^2 = 2\pi \sum\limits_{k=-\infty}^\infty |\hat{f}(k)|^2 = \lim\limits_{n\to\infty} ||f_n||_{L^2}^2 \overset{(\ast)}{=} \lim\limits_{n\to\infty} \left( ||f||_{L^2}^2 - ||f-f_n||^2_{L^2}\right) = ||f||_{L^2}^2$$ + $\implies \script{F}$ ist isometrisch und damit injektiv. \\ + Für $c\in l^2(\mathbb{C})$ ist $f_n = \sum\limits_{k=-n}^n c_k \text{e}^{-ikx}$ eine Cauchyfolge in $L^2(I,\mathbb{C})$ und konvergiert nach Riesz-Fischer gegen $f\in L^2(I,\mathbb{C}) \implies \script{F}(f) = c \implies \script{F}$ ist surjektiv. \end{proof} \begin{remark} @@ -330,7 +386,27 @@ \chapter{Konvergenzsätze und $L^n$-Räume} Im Fall $p = 1$ heißt eine Folge mit 1) und 2) \textbf{gleichgradig integrierbar}. \end{theorem} \begin{proof} - siehe Aufschrieb + \item[] \underline{a) $\implies$ b):} \\ + Sei $f_n \to f$ in $L^p(\mu)$. FÜr $A$ messbar gilt: + $$ \left| ||f_n||_{L^p(A)} - ||f||_{L^p(A)} \right| \leq ||f_n-f||_{L^p(A)} \leq ||f_n-f||_{L^p} \to 0$$ + $\implies \lim\limits_{n\to\infty} ||f_n||_{L^p(A)} = ||f||_{L^p(A)}$ und damit $\lambda(A) = \int\limits_{A}|f|^p d\mu$ \\ + 1) gilt dann nach Blatt 6, Aufgabe 2,3 \\ + (3) $\implies \lambda$ äußeres Maß, 2) $\implies$ Beh 1) \\ + Weiter ist mit Lemma IV.7 $\mu(E_\delta) \leq \frac{1}{\delta}\int |f|^p d\mu < \infty \text{für } E_\rho = \{ |f|^p \geq \delta \} $ \\ + $\implies \psi_{X\setminus E_\delta} |f|^p \leq \delta \to 0$ mit $\delta \to 0$, wobei $|f|^p$ integrierbare Majorante ist $\implies \lambda(X\setminus E_j) = \int\limits_{X\setminus E_j} |f|^p d\mu < \epsilon$ für $\delta$ klein genug. + \item[]\underline{b)$\implies$ a):}\\ + Zu $\epsilon > 0$ wähle $E$ wie in 2) sowie $\delta >0$ wie in 1). Da $\mu(E) < \infty$ existiert nach Egorov, Satz I.21 $A\subset E$ messbar mit $\mu(E\setminus A) < \delta$, sodass $f_n \to f$ gleichmäßig auf $A$. \\ + Hier wenden wir Satz I.21 auf das Maß $\mu_E = \mu|_{\script{D}(E)}$ an. + Die Menge $A$ ist dann $\mu_E$-messbar und damit auch $\mu$-messbar, denn $S \subset X$ und $E \mu$-messbar: + \begin{equation*} + \begin{split} \mu(S) &\geq \mu(S\cap E) + \mu(S\setminus E) \\ + &\geq \mu(S\cap A) + \mu (S\cap E \setminus A) + \mu(S\setminus E) \\ + &\geq \mu(S\cap A) + \mu(S\setminus A) + \end{split} + \end{equation*} + Nun gilt $|f_n -f|^p \leq \psi_A |f_n-f|^p + (\psi_{X\setminus E}+\psi_{E\setminus A}) 2^{p-1}(|f|^p+|f_n|^p)$ \\ + Lemma von Fatou impliziert für $B$ messbar $$\int\limits_B |f|^p d\mu \leq \liminf\limits_{n\to\infty} \int\limits_B |f_n|^p d\mu \leq \lambda(B)$$ + Mit $n\to\infty$ folgt aus 2) und 1) $$\limsup \int |f_n-f|^p d\mu \leq 2^p (\lambda(x\setminus E)+\lambda (E\setminus A)) < 2^{p+1}\epsilon$$ \end{proof} \ No newline at end of file diff --git a/chapter6.tex b/chapter6.tex index 66edca5..c5a05f7 100644 --- a/chapter6.tex +++ b/chapter6.tex @@ -35,14 +35,39 @@ \chapter{Satz von Fubini} $\alpha \times \beta$ ist ein äußeres Maß \end{lemma} \begin{proof} - siehe Aufschrieb + \item[$\cdot$] $\alpha \times \beta (\varnothing) \leq \alpha (\varnothing) \beta (\varnothing) = 0$ + \item[$\cdot$] Sei $E\subset \bigcup\limits_{i=1}^\infty E_i$ mit $E, E_i \subset X\times Y$. Wähle zu $\epsilon >0$ Überdeckungen $E_i \subset \bigcup\limits_{j=1}^\infty A_{i,j}\times B_{i,j}$ wie in $(\ast)$, sodass $\sum\limits_{j=1}^\infty \alpha (A_{i,k}) \beta(B_{i,j}) < \alpha \times \beta (E_i) + 2^{-i} \epsilon$ \\ + $\implies E\subset \bigcup\limits_{i,j=1}^\infty A_{i,j}\times B_{i,j}$\\ + $\implies \alpha \times \beta (E) \leq \sum\limits_{i,j=1}^\infty \alpha(A_{i,j})\beta(B_{i,j}) < \sum\limits_{i=1}^\infty \alpha \times \beta (E_i)+\epsilon$ \end{proof} \begin{lemma} Sei $P = A \times B$. Eine Produktmenge, d.h. $A,B$ sind messbar bzgl. $\alpha$ bzw. $\beta$. Dann gilt $\alpha \times \beta(P) = \alpha(A) \beta(B)$ und $P$ ist $\alpha \times \beta$-messbar. \end{lemma} + \begin{proof} - siehe Aufschrieb + $A,B$ messbar $\implies alpha\times\beta(P)\overset{(\ast)}{\leq}\alpha(A)\beta(B)$\\ + \item[]\underline{z.z:} $\alpha (A) \beta(B) \leq \alpha\times\beta(P)$\\ + Der $y$-Schnitt von $P$ ist $P_y = \{x\in X: (x,y) \in P \} = \begin{cases} A \text{, für } y\in B \\ + \varnothing \text{ sonst } \end{cases}$ \\ + Ist $P \subset \bigcup\limits_{j=1}^\infty P_j$ mit $P_j = A_j\times B_j$, so folgt $P_y \subset \bigcup\limits_{j=1}^\infty (P_j)_y$ und mit Fatou \\ $\alpha(A)\beta(B) = \int\limits_y \alpha(P_y) d\beta(y) \leq \int\limits_y \sum\limits_{j=1}^\infty \alpha((P_j)_y) d\beta(y) = \sum\limits_{j=1}^\infty \int\limits_y \alpha((P_j)_y) d\beta(y) = \sum\limits_{j=1}^\infty \alpha(A_j)\beta(B_j)$ \\ + $\implies \alpha(A)\beta(B) \leq \alpha\times\beta(P)$\\ + \item[]Messbarkeit: \\ + Sei $S\subset\bigcup\limits_{j=1}^\infty P_k$ mit $P_j = A_j\times B_j$ wie in $(\ast)$. \\ + Es gilt: + \begin{align*} + (\ast\ast) && P_j \cap P = (A_j \cap A) \times (B_j \cap B) && P_j\setminus P = (A_j\cap A) \times (B_j\setminus B) \cup (A_j\setmiinus A) \times B_j + \end{align*} + $\implies$ + \begin{equation*} + \begin{split} + \alpha\times\beta(P_j\cap P) + \alpha\times\beta (P_j\setminus P) &\leq \alpha(A_j \cap A) \beta(B_j\cap B) + \alpha(A_j\cap A)\beta(B_j\setminus B) + \alpha(A_j\setminus A) \beta(B_j) \\ + &= \alpha(A_j\cap A) \beta(B_j) + \alpha(A_j\setminus A)\beta(B_j) \\ + &= \alpha(A_j) \beta(B_j) + \end{split} + \end{equation*} + $\implies \alpha\times\beta(S\cap P) + \alpha\times\beta(S\setminus P) \leq \sum\limits_{j=1}^\infty \alpha\times\beta(P_j\cap P) + \sum\limits_{j=1}^\infty \alpha\times\beta (P_j \setminus P) \leq \sum\limits_{j=1}^\infty \alpha(A_j)\beta(B_j)$\\ + $\overset{(\ast)}{\implies} \alpha\times\beta(S\cap P) + \alpha\times\beta(S\setminus P) \leq \alpha\times\beta(S)$ \end{proof} 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betrachte induktiv $P^r\setminus \bigcup\limits_{j=1}^{i-1}P_j^r$. Nach $(\ast\ast)$ ist dies eine disjunkte Vereinigung von Produktmengen. + \item[b)] $E^r\subset E^{r-1}$ $\forall r$, sonst gehe über zur disjunkten Überdeckung $P^r_i \cap P_j^{r-1}$ mit $i,j\in\mathbb{N}$. Nach $(\ast\ast)$ sind dies Produktmengen und \\ + $\sum\limits_{i,j=1}^\infty \alpha\times\beta (P_i^r\cap P_j^{r=1}) = \sum\limits_{i=1}^\infty \alpha\times\beta(\bigcup\limits_{j=1}^\infty P_i^r\cap P_j^{r-1}) \leq \sum\limits_{i=1}^\infty\alpha\times\beta (P_i^r)<\alpha\times\beta (D) + \frac{1}{r}$ Mit $E:=\bigcap\limits_{r=1}^\infty E^r$ folgt: $D\subset \bigcap\limits_{r=1}^\infty E^r = E$ und $\alpha\times\beta(E) = \alpha\times\beta(D)$. Da $D$ messbar ist, gilt $\alpha\times\beta(E\setminus D) = \alpha\times\beta (E) - \alpha\times\beta(D) = 0$. + \end{itemize} + \item[]\underline{Wir wollen 3 Aussagen zeigen} + \begin{itemize} + \item[1)] $E_y = \{x\in X: (x,y)\in E \}$ ist $\alpha$-messbar $\forall y\in Y$ + \item[2)] $f_k: Y \to [0,\infty]$, $y \mapsto \alpha(E_y)$ $\beta$-messbar + \item[3)] $\gamma (?) (E) := \int\limits_Y f_k d\beta = \alpha\times\beta (E)$ + \end{itemize} + Sei $\epsilon$ das System aller $\alpha\times\beta$-messbaren Mengen, welche 1), 2) und 3) erfüllen. Für Produktmenten $A\times B$ gilt: \\ + $(A\times B)_y = A$ falls $y=B$ und $(A\times B)_y = \varnothing$, falls $y\notin B$ $\implies A\times B \in \epsilon$, denn $f_{A\times B} = \alpha (A)\psi_B$ und $\gamma(A\times B) = \alpha(A) \beta(B) = \alpha\times\beta(A\times B)$. \\ + Jetzt betrachte disjunkte Vereinigung $E = \bigcup\limits_{i=1}^\infty E_i$ mit $E_i \in \epsilon$ $\forall i\in \mathbb{N}$. \\ + $\implies E_y = \bigcup\limits_{i=1}^\infty (E_i)_y$ $\alpha$-messbar \\ + $f_E = \sum\limits_{i=1}f_{E_i}$ ist $\beta$-messbar und \\ + $\gamma(E) = \int\limits_Y \sum\limits_{i=1}^\infty f_{E_i} d\beta \overset{\text{Satz von der Mon. Konv.}}{=} \sum\limits_{i=1}^\infty \int\limits_Y f_{E_i} d\beta = \sum\limits_{i=1}^\infty \alpha\times\beta(E_i) = \alpha\times\beta(E)$ \\ + Schließlich sei $E^1 \supset E^2 \supset ...$ mit $E^r \in \epsilon$ und $\alpha\times\beta(E^1) < \infty$. \\ + Für $E = \bigcap\limits_{r=1}^\infty E^r$ ist $E_y = \bigcap_{r=1}^\infty (E^r)_y$ $\alpha$-messbar $\forall y\in Y$. Lemma II.8 impliziert \\ + $f_E(y) = \alpha (E_y) = \lim\limits_{r\to\infty} \alpha((E^r)_y) = \lim\limits_{r\to\infty} f_{E^r}(y)$ für $\beta$-fast alle $y\in Y$. \\ + $\implies f_E$ ist $\beta$-messbar. \\ + Zuletzt folgt aus Satz von Lebesgue wegen $f_{E^r} \leq f_{E^1}$ \\ + $\gamma(E) = \int\limits_Y f_E d\beta = \lim\limits_{r\to\infty} \int\limits_Y f_{E^r} d\beta = \lim\limits_{r\to\infty}\alpha\times\beta(E^r) = \alpha\times\beta(E)$ \\ + $\implies$ Für $E$ wie oben gelten somit die Aussagen 1), 2), und 3). \\ + Jetzt wende das Argument auf eine $\alpha\times\beta$ Nullmenge $N$ statt $D$ an. Erhalte $C\supset N$ mit $C\in\epsilon$ und $\alpha\times\beta(C) = 0$. Also \\ + $0 = \alpha\times\beta(C) = \int\limits_Y \alpha(C_y)d\beta(y) \implies \alpha(N_y) \leq \alpha(C_y) = 0$ für fast alle $y\in Y$. \\ + Wähle $N=E\setminus D$. Dann folgt für $\beta$-fast alle $y\in Y$: \\ + $D_y = E_y \setminus N_y$ ist $\alpha$-messbar und es gilt: $f_D(y) = f_E(y)$ ($\implies f_D$ ist $\beta$-messbar)\\ + $\int\limits_Y f_D d\beta = \int\limits_Y f_E d\beta = \alpha\times\beta(E) = \alpha\times\beta(D)$. + \item[II] Sei $D\subset X\times Y$ nur messbar.\\ + Nach Vor. gilt $X=\bigcup\limits_{n=1}^\infty A_n$, $Y=\bigcup\limits_{n=1}^\infty B_n$ mit $A_n$, $B_n$ messbar und $\alpha(A_n)$, $\beta(B_n) < \infty$. \\ + O.E. $A_n$, $B_n$ aufsteigend. $D_n = D\cap (A_n\times B_n)$ ist $\alpha\times\beta$-messbar mit $(\alpha\times\beta)(D_n) < \infty$ \\ + $\overset{\text{I}}{\implies}$ Dann ist $D_y = \bigcup\limits_{n=1}^\infty (D_n)_y$ $\alpha$-messbar für $\beta$-fast alle y. Es gilt $f_{D_1} \leq f_{D_2} \leq ...$ und $f_D(y) = \alpha(D_y) = \lim\limits_{n\to\infty}(\alpha(D_n)_y) = \lim\limits_{n\to\infty}f_{D_n}(y) \\ + \implies f_D$ $\beta$-messbar und $\int\limits_Y f_D d\beta \overset{\text{mon.konv.}}{=} \lim\limits_{n\to\infty}\int\limits_Y f_{D_n}d\beta = \lim\limits_{n\to\infty} \alpha\times\beta(D_n) = \alpha\times\beta(D)$ \end{proof} \begin{remark} @@ -97,14 +133,26 @@ \chapter{Satz von Fubini} Mit $I_k = [\frac{k-1}{n}, \frac{k}{n}]$ gilt $D = \bigcap\limits_{n=1}^{\infty}( \bigcup\limits_{k=1}^{\infty} I_k \times I_k) \implies D$ ist messbar bzgl. $\lambda^1 \times card$ \end{example} - \newpage \begin{lemma} Es gilt $\lambda^n = \lambda^k \times \lambda^m$ für $k + m = n$ \end{lemma} + \begin{proof} - siehe Aufschrieb + Quader im $\mathbb{R}^n$ sind Produkte $P\times Q$ von Quadern im $\mathbb{R}^k$ und $\mathbb{R}^m$ und es gilt $vol^n(P\times Q) = vol^k(P)vol^m(Q)$. Somit ist $\lambda^n(E) = \inf \{\sum\limits_{j=1}^\infty vol^k(P_j)vol^m(Q_j):$ $ P_j, Q_j$ Quader, $E\subset \bigcup\limits_{j=1}^\infty P_j\times Q_j \}$ \\ + Es folgt direkt $\lambda^k \times \lambda^m \leq \lambda^n$. \\ + \item[\underline{z.z}] $\lambda^n \leq \lambda^k\times\lambda^m$ \\ + Reicht dies für Produktmengen zu zeigen, d.h. $\lambda^n(A\times B) \leq \lambda^k(A) \lambda^m(B)$, denn daraus folgt für $E\subset \mathbb{R}^n$ bel.: + $$ \lambda^n(E) \leq \inf\{\sum\limits_{j=1}^\infty \lambda^n(A_j\times B_j): A_j, B_j \text{ messbar }, E\subset \bigcup\limits_{j=1}^n A_j\times B_j \} \leq (\lambda^k\times\lambda^m)(E)$$ + Betrachte nun $A\times B$ mit $\lambda^k(A)$, $\lambda^m(B) <\infty$. Zu $\epsilon >0$ existiert Quader $P_i$, $Q_i$ mit $A\subset \bigcup\limits_{i=1}^\infty P_i$, $B\subset \bigcup\limits_{j=1}^\infty Q_j$, sodass $\sum\limits_{i=1}^\infty vol^k(P_i) < \lambda^k(A)+\epsilon$, $\sum\limits_{j=1}^\infty vol^m(Q_j) < \lambda^m(B)+\epsilon$. \\ + $\implies A\times B \subset \bigcup\limits_{i,j=1}^\infty (P_i\times Q_j) \\ + \implies \lambda^n(A\times B) \leq \sum\limits_{i,j=1}^\infty vol^k(P_i) vol^m(Q_j) < (\lambda^k(A) + \epsilon) (\lambda^m(B) + \epsilon) \overset{\epsilon\to 0}{\to} \lambda^k(A)\lambda^m(B)$. \\ + Für $\lambda^k(A)\lambda^m(B) = \infty$ ist nichts zu zeigen. \\ + Bleibt der Fall $\lambda^k(A) = \infty$, $\lambda^m(B) = 0$ (oder umgekehrt).\\ + Mit $A_l = \{x\in A:$ $l-1\leq ||x|| \leq l \}$ folgt \\ + $$\lambda^n(A\times B) \leq \sum\limits_{l=1}^\infty \lambda^n(A_l\times B) \leq \sum\limits_{l=1}^\infty \lambda^k(A_l)\lambda^m(B) = 0$$ \end{proof} - + \newpage + \sidenote{Vorlesung 18}{15.01.21} \begin{example} \begin{enumerate} @@ -149,7 +197,38 @@ \chapter{Satz von Fubini} \end{theorem} \begin{proof} - siehe Aufschrieb + Für $f = \psi_E$ mit $E\subset X\times Y$ $\sigma$-endlich gelten 1)-3) nach Satz VI.4 + \begin{itemize} + \item[$\cdot$] Für $\beta$-fast alle $y\in Y$ ist $f(\cdot, y) = \psi_{E_y}$ $\alpha$-messbar mit $\int\limits_X f(x,y) d\alpha(x) = \alpha(E_y)$ + \item[$\cdot$] $y \mapsto \alpha(E_y)$ ist $\beta$-messbar mit $\int\limits_Y \alpha(E_y) d\beta(y)$ + \item[$\cdot$] $\int\limits_{X\times Y} f d(\alpha\times\beta) = (\alpha\times\beta)(E) = \int\limits_Y \alpha(E_y)d\beta(y) = \int\limits_Y \int\limits_X f(x,y) d\alpha(x) d\beta(y)$ + \end{itemize} + Sei jetzt $f\leq 0$. Satz IV.9 $\implies \exists \alpha\times\beta$-Treppenfunktion $0 \leq f_1 \leq f_2 \leq ...$ mit $f_k(x,y) \to f(x,y)$ auf $X\times Y$. Die $f_k$ sind ebenfalls $\sigma$-endlich, und es gilt: + \begin{itemize} + \item[$\cdot$] $f(\cdot, y)$ ist monotoner Grenzwert der $f_k(\cdot, y)$. Für $\beta$-fast alle $y\in Y$ ist damit $f(\cdot, y)$ $\alpha$-messbar und $\int\limits_X f(\cdot, y)d\alpha = \lim\limits_{k\to\infty} \int\limits_x f_k(\cdot, y) d\alpha$ (Satz IV.10) + \item[$\cdot$] $y \mapsto \int\limits_X f(\cdot, y) d\alpha$ ist monotoner Grenzwert von $y\mapsto \int\limits_X f_k(\cdot, y) d\alpha$ für $\beta$-fast alle $y\in Y$\\ $\implies y\mapsto \int\limits_X f(\cdot, y) d\alpha$ ist $\beta$-messbar und $\int\limits_Y \int\limits_X f(x,y)d\alpha(x) d\beta(x) = \lim\limits_{k\to\infty}\int\limits_Y \int\limits_X f_k(x,y) d\alpha(x) d\beta(y)$ + \item[$\cdot$] $\int\limits_{X\times Y} f d(\alpha\times\beta) = \lim\limits_{k\to\infty}\int\limits_{X\times Y} f_k d(\alpha\times\beta) = \lim\limits_{k\to\infty}\int\limits_Y\int\limits_X f_k (x,y) d\alpha(x) d\beta(y) = \int\limits_Y \int\limits_X f(x,y) d\alpha(x) d\beta(y) $ + \end{itemize} + $\implies$ Fubini für $f\geq 0$ \\ + Sei $f$ nun $\sigma$-endlich und $\int f^- d(\alpha\times\beta) < \infty$. \\ + $\implies f^\pm$ sind auch $\sigma$-endlich und $\int\limits_Y \int\limits_X f^-(x,y) d\alpha(x) d\beta(y) = \int\limits-{X\times Y} f^- d(\alpha\times\beta) y\infty$ \\ + $\implies$ Für $\beta$-fast alle $y\in Y$ gilt $\int\limits_X f^-(x,y) d\alpha(x) < \infty$ und $f^-(x,y) < \infty$ für $\alpha$-fast alle $x\in X$. + \begin{itemize} + \item[$\cdot$] Für $\beta$-fast alle $y\in Y$ ist $f(\cdot, y) = f^+(\cdot, y) - f^-(\cdot, y)$ $\alpha$-messbar, mit Integral $\int\limits_X f(\cdot, y) d\alpha = \int\limits_X f^+(\cdot,y)d\alpha - \int\limits_X f^-(\cdot,y)d\alpha$ + \item[$\cdot$] $y\mapsto \int\limits_X f(\cdot,y) d\alpha$ ist Differenz zweier messbarer Funktionen, also messbar und ihr Integral existiert, denn $s\mapsto s^-$ fallend, gilt \\ + $\int\limits_Y \left(\int\limits_X f(x,y) d\alpha(x) \right)^- d\beta(y) \leq \int\limits_Y \left(-\int\limits_X f^-(x,y) d\alpha(x) \right)^- d\beta(y) \\= \int\limits_Y \int\limits_X f^-(x,y) d\alpha(x) d\beta(y) < \infty$ + \item[$\cdot$] \begin{equation*} + \begin{split} + \int\limits_{X\times Y} f d(\alpha\times\beta) &= \int\limits_{X\times Y} f^+ d(\alpha\times\beta) - \int\limits_{X\times Y} f^- d(\alpha\times\beta) \\ + &= \int\limits_Y \int\limits_X f^+(x,y) d\alpha(x) d\beta(y) - \int\limits_Y \int\limits_X f^-(x,y) d\alpha(x) d\beta(y) \\ + &= \int\limits_Y \int\limits_X \left( f^+(x,y) -f^-(x,y)\right) d\alpha(x) d\beta(y) \\ + &= \int\limits_Y \int\limits_X f(x,y) d\alpha(x) d\beta(y) + \end{split} + \end{equation*} + $\implies$ Beh. + \end{itemize} + Zusatz: Folgt durch Anwendung auf $|f|$, dann gilt: $$ \int\limits_{X\times Y} |f| d(\alpha\times\beta) = \int\limits_Y \int\limits_X |f(x,y)|d\alpha(x) d\beta(y) < \infty$$ + $\implies$ Integral von $f$ bzgl $\alpha\times\beta$ ist definiert. \end{proof} \begin{example} @@ -186,7 +265,19 @@ \chapter{Satz von Fubini} $$\int\limits_{\Omega}(\partial_j f)g dx = -\int\limits_{\Omega} f (\partial_j g) dx \ \ \ \ \forall \ 1 \leq j \leq n \ (dx \ \hat{=} \ d \lambda^n)$$ \end{theorem} \begin{proof} - siehe Aufschrieb + Es reicht die Aussage $(\ast)$ zu zeigen: + \begin{align*} + (\ast) && \int\limits_{\mathbb{R}^n} \partial_j f dx = 0 \text{ } \forall f\in C_c^1 (\mathbb{R}^n) + \end{align*} + Denn setzen wir $fg$ durch $0$ zu einer Funktion $\phi \in C^1_c(\mathbb{R}^n)$ fort, so folgt $$ 0 = \int\limits_{\mathbb{R}^n} \partial_j \phi dx = \int\limits_\Omega \partial_j (fg) dx = \int\limits_\Omega (\partial_j f) g dx + \int\limits_\Omega f \partial_j g dx$$ + Fubini für $f\in C^1_c(\mathbb{R}^n)$ liefert mit $x = (x', x_n)\in \mathbb{R}^{n-1}\times \mathbb{R}$ + \begin{equation*} + \begin{split} + \int\limits_{\mathbb{R}^n} \partial_j f(x) dx &= \int\limits_{\mathbb{R}} \int\limits_{\mathbb{R}^{n-1}} \partial_j f(x',x_n) dx' dx_n \\ + &= \int\limits_{\mathbb{R}^{n-1}} \int\limits_{\mathbb{R}} \partial_j f(x',x_n) dx_n dx' + \end{split} + \end{equation*} + Für $j=n$ ist das letzte Integral $0$ nach Hauptsatz. Für $1 \leq j \leq n-1$ verschwindet das mittlere Integral nach Induktion. \end{proof} \begin{remark} From ef1e5f341fba1a5a580baac6408192af41035389 Mon Sep 17 00:00:00 2001 From: Liz Date: Thu, 25 Feb 2021 17:02:50 +0100 Subject: [PATCH 6/6] bis auch Beweise chapter10 --- analysis_3_skript.pdf | Bin 677751 -> 733561 bytes analysis_3_skript.tex | 1 + chapter10.tex | 117 ++++++++++++++++++ chapter7.tex | 88 +++++++++++-- chapter8.tex | 88 +++++++++++-- chapter9.tex | 66 ++++++++-- .../IX_1_Plaettung.png | Bin 7 files changed, 332 insertions(+), 28 deletions(-) create mode 100644 chapter10.tex rename "img/IX_1_Pl\303\244ttung.png" => img/IX_1_Plaettung.png (100%) diff --git a/analysis_3_skript.pdf b/analysis_3_skript.pdf index 54466ca9ee6a10e80f67abf83bfbb89dcd87d910..c48ac921ccc96a95db01724e0b0de7bb9e36e162 100644 GIT binary patch delta 263359 zcmV(~K+nJTt||HAEP#XogaU*Ev;^~r0yi;}fzT;`&00y5Q%4fM_pf-|)L~TAzQA}Q zhG`&hI6Q;h=3;yh^y@~A7A&c4;4t&oC#&98Rm!)tkt5~+TDp{#mGx!jmzj?iHw7(T 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--- /dev/null +++ b/chapter10.tex @@ -0,0 +1,117 @@ +\chapter{Faltung und Fouriertransformation} +\begin{theorem} + Sei $f\in L^p(\mathbb{R}^n)$, $1\leq p \leq \infty$ und $g\in L^1(\mathbb{R}^n)$. \\ + Die Faltung von $f$ mit $g$ ist die $\lambda^n$-fast überall definierte Funktion + $$ f\ast g: \mathbb{R}^n \to \mathbb{\bar{R}}\text{, } (f\ast g)(x) = \int\limits_{\mathbb{R}^n} f(x-y) g(y) dy $$ + Es gilt $f\ast g\in L^p(\mathbb{R}^n)$ mit $||f\ast g||_{L^p} \leq ||f||_{L^p} ||g||_{L^1}$ +\end{theorem} +\begin{proof} +Aufschrieb +\end{proof} + +\begin{lemma} + $f\in L^p(\mathbb{R}^n)$ mit $1\leq p \leq \infty$ und $\tau_h: \mathbb{R}^n\to \mathbb{R}^n$, $\tau_h(x)=x+h$. Dann gelten: + \item[i)] $f\circ \tau_h \in L^p(\mathbb{R}^n)$ mit $||f\circ \tau_h ||_{L^p} = ||f||_{L^p}$ + \item[ii)] $f\circ \tau_h \to f$ in $L^p(\mathbb{R}^n)$ für $h\to 0$, falls $1\leq p < \infty$ +\end{lemma} +\begin{proof} + \item[i)] Trivial + \item[ii)] s. Blatt 9, Aufgabe 1 +\end{proof} + +\begin{theorem} + $f\in L^p(\mathbb{R}^n)$ mit $ 1\leq p \leq \infty$. Ist $\eta \in L^1(\mathbb{R}^n)$ mist $\int\limits_{\mathbb{R}^n} \eta(z) dz = 1$, so folgt $\eta_\rho (x) = \rho^{-n} \eta(\frac{x}{\rho})$: + \begin{equation*} + ||f\ast \eta_\rho||_{L^p} \leq ||f||_{L^p} ||\eta||_{L^1} \\ + \text{ und } \\ + f\ast \eta_\rho \to f \text{ in } L^p(\mathbb{R}^n) \text{ für } \rho \to 0 + \end{equation*} +\end{theorem} + +\begin{proof} + siehe Aufschrieb +\end{proof} + +\begin{theorem} + Sei $\eta \in C^k(\mathbb{R}^n)$ mit $|| D^\alpha \eta ||_{C^0(\mathbb{R}^n)} \leq C_k$ für $|\alpha| \leq k$ $(\alpha = (\alpha_1, ..., \alpha_n), |\alpha| = \alpha_1+...+\alpha_n, D^\alpha = \partial_1^{\alpha_1}\cdot ... \cdot \partial_n^{\alpha_n})$. Für $f \in L^1(\mathbb{R}^n)$ ist dann $f\ast \eta \in C^k(\mathbb{R}^n)$ und es gilt $D^\alpha (f\ast \eta) = f\ast (D^\alpha \eta)$, speziell $||D^\alpha(f\ast \eta)||_{C^0(\mathbb{R}^n)} \leq C_k ||f||_{L^1}$ +\end{theorem} +\begin{proof} + siehe Aufschrieb +\end{proof} + +\begin{theorem} + $\Omega \subset\mathbb{R}^n$ offen und $1\leq p < \infty$. Dann ex. zu $f\in L^p(\Omega)$ eine Folge $f_k \in C^\infty_C(\Omega)$ mit $||f-f_k||_{L^p}\to 0$ mit $k \to \infty$. +\end{theorem} +\begin{proof} + siehe Aufschrieb +\end{proof} + +\begin{definition} + Die Fourier-Transformierte von $f\in L^1(\mathbb{R}^n, \mathbb{C})$ ist die Funktion + $$\hat{f}: \mathbb{R}^n \to \mathbb{C}\text{, } \hat{f}(p) = (2\pi)^{-\frac{n}{2}} \int\limits_{\mathbb{R}^n} f(x) \text{e}^{-i}dx$$ + Die Inverse Fourier-Transformierte von $g\in L^1(\mathbb{R}^n, \mathbb{C})$ ist $$ + \check{g}:\mathbb{R}^n \to \mathbb{C}\text{, } \check{g}(x) = (2\pi)^{-\frac{n}{2}} \int\limits_{\mathbb{R}^n} g(p) \text{e}^{i}dp$$ +\end{definition} + +\begin{theorem} + Für $f,g\in L^1(\mathbb{R}^n,\mathbb{C})$ gilt: + \item[1)] $\hat{f}\in C^0(\mathbb{R}^n,\mathbb{C})$ und $||\hat{f}||_{C^0(\mathbb{R}^n)} \leq (2\pi)^{-\frac{n}{2}}||f||_{L^1}$ + \item[2)] $\widehat{f\ast g} = (2\pi)^{\frac{n}{2}}\hat{f}\hat{g}$ + \item[3)] $<\hat{f},g>_{L^2(\mathbb{R}^n)} = < f,\check{g}>_{L^2(\mathbb{R}^n)}$ +\end{theorem} +\begin{proof} + siehe Aufschrieb +\end{proof} + +\begin{example} + siehe Aufschrieb +\end{example} + +\begin{theorem}[Plancherel] + Für $f\in (L^1\cap L^2)(\mathbb{R}^n,\mathbb{C})$ gilt $||\hat{f}||_{L^2} = ||\check{f}||_{L^2} = ||f||_{L^2}$ +\end{theorem} + +\begin{proof} + siehe Aufschrieb +\end{proof} + +\begin{theorem} + Es gibt eine eindeutig bestimmte Abb. $\script{F}, \script{F}^\ast: L^2(\mathbb{R}^n,\mathbb{C}) \to L^2(\mathbb{R}^n,\mathbb{C})$ mit + \item[1] $\script{F}f = \hat{f}$, $\script{F}^\ast f = \check{f}$ $\forall f\in (L^1\cap L^2)(\mathbb{R}^n,\mathbb{C})$ + \item[2] $||\script{F}f||_{L^2} = ||f||_{L^2} = ||\script{F}^\ast f||_{L^2}$ $\forall f\in L^2(\mathbb{R}^n,\mathbb{C})$ + Weiter gelten für $f,g\in L^2(\mathbb{R}^n,\mathbb{C})$ + \item[3] $<\script{F}f,\script{F}g>_{L^2} = _{L^2} = <\script{F}^\ast f, \script{F}^\ast g>_{L^2}$ + \item[4] $<\script{F}f,g> _{L^2} = _{L^2}$ + \item[5] $\script{F}^\ast \script{F} = \script{F} \script{F}^\ast = \text{Id}_{L^2(\mathbb{R}^n)}$ +\end{theorem} +\begin{proof} + siehe Aufschrieb +\end{proof} + +\begin{notation} + $\script{F}f = \hat{f}$, $\script{F}^\ast f = \check{f}$ auch wenn $f,g$ nur in $L^2$ +\end{notation} + +\begin{definition}[Schwartz-Raum] + $S(\mathbb{R}^n,\mathbb{C}) = \{f\in C^\infty (\mathbb{R}^n,\mathbb{C}): x^\alpha D^\beta f$ ist beschränkt für alle $\alpha,\beta \in \mathbb{N}^n_0 \}$ \\ + $(x^\alpha = x_1^{\alpha_1}... x_n^{\alpha_n}), D^\beta = \partial_1^{\beta_1}...\partial_n^{\beta_n}$ +\end{definition} + +\begin{remark} + $f\in S(\mathbb{R}^n,\mathbb{C}) \implies |f(x)| \leq c_N (1+||x||^2)^{-\frac{N}{2}}$ $\forall N\in \mathbb{N}$ \\ + $\implies f\in L^p(\mathbb{R}^n,\mathbb{C})$ $\forall p\in [1,\infty]$ \\ + Außerdem: $f\in S(\mathbb{R}^n,\mathbb{C}) \implies \partial_j f$ und $x_jf \in S(\mathbb{R}^n,\mathbb{C})$ $\forall 1\leq j \leq n$ +\end{remark} + +\begin{theorem} + Mit $f$ ist auch $\hat{f}\in S(\mathbb{R}^n,\mathbb{C})$ und + \item[1)] $\widehat{\partial_j f}(p) = i p_j \hat{f}(p)$ sowie $\widehat{x_j f}(p) = i \partial_j \hat{f}(p)$ + \item[2)] $f(x) = (2\pi)^{-\frac{n}{2}}\int\limits_{\mathbb{R}^n} \hat{f}(p) \text{e}^{i}dp$ $\forall x\in \mathbb{R}^n$ +\end{theorem} +\begin{proof} + siehe Aufschrieb +\end{proof} + +\begin{example} + siehe Aufschrieb +\end{example} \ No newline at end of file diff --git a/chapter7.tex b/chapter7.tex index a5f4e9b..dc60df4 100644 --- a/chapter7.tex +++ b/chapter7.tex @@ -33,9 +33,26 @@ \chapter{Der Transformationssatz} $$\limsup\limits_{j \to \infty} \frac{\lambda^n(\Phi(Q_j))}{\lambda^n(Q_j)} \leq |det D\Phi(x_0)|$$ \end{lemma} \begin{proof} - siehe Aufschrieb + O.E. $x_0 = 0$, $\phi(0) = 0$ (sonst betrachte $\tilde{\phi}(x) = \phi(x+x_0) - \phi (x_0)$) + \item[]\underline{I. $D\phi (0) = \text{Id}$} \\ + $\overset{\phi \text{ diff in } 0}{\implies} 0 = \lim\limits_{x\to 0} \frac{||\phi(x) - (\phi (0) + D\phi (0) x) ||_\infty}{||x||_\infty} = \lim\limits_{x\to 0} \frac{|| \phi(x) - x ||_\infty}{||x||_\infty}$ \\ + (wegen $||x||_\infty \leq ||x|| \leq \sqrt{n} ||x||_\infty$ darf die $|| \cdot ||_\infty$ benutzt werden). \\ + Sei nun $\epsilon > 0$ für $x\in Q_j$ gilt: + $$ ||x||_\infty \leq ||x-x_j||_\infty + ||x_j-0||_\infty \leq 2 \phi_j$$ + $\implies$ Für $j$ hinreichend groß gilt: \\ + $$||\phi (x) - x ||_\infty \leq \epsilon ||x||_\infty \leq 2\epsilon \phi_j \text{ } \forall x\in Q_j$$ + $\implies ||\phi(x) - \phi(x_j)||_\infty \leq ||\phi(x) - x||_\infty + ||x-x_j||_\infty + ||\phi(x_j) - x_j||_\infty \leq (1+4\epsilon)\phi_j$ $\forall x\in Q_j$ \\ + $\implies \phi (Q_j) \subset Q(\phi(x_j), (1+4\epsilon)\phi_j)$ \\ + $\implies \frac{\lambda^n(\phi(Q_j))}{\lambda^n (Q_j)} \leq (1+4\epsilon)^n$ \\ + Mit $j\to\infty$ und $\epsilon\to\infty$ folgt die Behauptung im Fall I. + \item[] \underline{II $D\phi(0) \in GL_n(\mathbb{R})$} \\ + Sei $S = D\phi(0)$ und $\phi_0 := S^{-1} \circ \phi$, also $D\phi_0(0) = \text{Id}$. + \begin{align*} + \limsup\limits_{j\to\infty} \frac{\lambda^n(\phi(Q_j))}{\lambda^n(Q_j)} &= \limsup\limits_{j\to\infty} \frac{\lambda^n(S(\phi_0(Q_j)))}{\lambda^n(Q_j)} \\ + &= |\det{S}| \limsup\limits_{j\to\infty} \frac{\lambda^n(\phi_0(Q_j))}{\lambda^n(Q_j)} \\ + &\leq |\det{S}| = \det{D\phi(0)} + \end{align*} \end{proof} - \begin{theorem}[Transformationsformel] $\script{U}, \script{V} \subseteq\mathbb{R}^n$ offen, $\Phi:\script{U} \to \script{V} \ C^1$-Diffeomorphismus. Ist $A \subseteq \script{U}$ $\lambda^n$-messbar, so ist auch $\Phi(A)$ $\lambda^n$-messbar und es gilt: \begin{enumerate} @@ -49,7 +66,51 @@ \chapter{Der Transformationssatz} \end{theorem} \begin{proof} - siehe Aufschrieb + Betrachte auf $\sigma$-Algebra $\script{A}$ der $\lambda^n$-messbaren Mengen $A\subset U$ die Maße + \begin{align*} + \lambda(A) = \int\limits_A |\det{D\phi}|d\lambda^n && \mu(A) = \lambda^n(\phi(A)) + \end{align*} + Def. Bildmaß $\rightarrow \mu(A) = \psi(\lambda^n)(A)$ mit $\psi = \phi^{-1}$. \\ + Für $\lambda^n$-messbares $A\subset U$ ist $\psi^{-1}(A) = \phi(A)$ $\lambda^n$-messbar (Satz III.13) und damit ist $A$ auch $\mu$-messbar (s.Blatt 2, A4). \\ + Die Einschränkung des äußeren Maßes $\mu$ auf $\script{A}$ ist somit ein Maß. \\ + Weiter ist $\lambda$ das Lebesguemaß mit Dichte $|\det{D\phi}|$ (s.Blatt 6, A3), und damit folgt die Maßeigenschaft von $\lambda$ aus dieser Aufgabe. \\ + Weiter folgt aus Blatt 6, A3 bzw. Satz III.13 $(\ast)$ $N\subset U$ mit $\lambda^n(N)= 0 \implies \lambda(N) = \mu(N) = 0$ + \item[]\underline{z.z: $\lambda \geq \mu$} \\ + O.E. sei $\lambda(U) < \infty$, $\mu(U) < \infty$. Sonst schöpfe $U$ durch offene, relativ kompakte Mengen aus. \\ + Satz III.7 $\implies$ Jede $\lambda^n$-messbare Menge $A$ ist von der Form $A = E \setminus N$ mit $\lambda^n(N) = 0$ und $E = \bigcap\limits_{i=1}^\infty U_i$ mit offenen $U_1 \supset U_2 \supset ...$ \\ + Satz I.7 (ii) $\implies$ Reicht aus $\lambda \geq \mu$ auf allen offenen Mengen nachzuweisen. \\ + Lemma III.3 $\implies$ Jede offene menge ist abzählbare Vereinigung von kompakten, achsenparallelen Würfeln in $U$ mit paarweise disjunktem Inneren. \\ + Damit bleibt zu zeigen: $$ \lambda(Q_0) \geq \mu(Q_0) \text{ } \forall Q_0 = Q(r_0, \rho_0)\subset U$$ + Angenommen es ist $\lambda(Q_0) < \mu(Q_0)$ für ein $Q_0 = Q(r_0, \rho_0)$. Wähle $\theta \in [0,1]$ mit $\lambda(Q_0 < \theta \mu(Q_0))$ und zerlege $Q_0$ durch Halbierung der Kanten in $2^n$ kompakte Teilwürfel $Q_{0,1}, Q_{0,2},...,Q_{0,2^n}$. Wähle $\lambda(Q_{0,1}) \geq \theta \mu(Q_{0,1})$ $\forall i \implies \lambda(Q_0) = \sum\limits_{i=1}^{2^n} \lambda{Q_{0,1}} \geq \theta \sum\limits_{i=1}^{2^n} \mu(Q_{0,1}) = \theta \mu(Q_0)$ (Widerspruch !!) \\ + (Ränder sind Nullmengen bzgl $\lambda, \mu$ nach $(\ast)$) \\ + $\implies$ Unter $Q_{0,1}$ ex. einer (nennen wir $Q_1$) mit $\lambda(Q_1) < \theta \mu(Q_i)$ \\ + Bestimme damit induktiv eine Schachtelung $Q_0 \supset Q_1 \supset ...$ mit $(\ast\ast)$ $\lambda(Q_j) < \theta \mu(Q_j)$ $\forall j\in\mathbb{N}$ \\ + Es gilt $\bigcap\limits_{j=1}^\infty Q_j = \{x_0\}$ mit $x_0 \in U$ \\ + Da $\det{D\phi}$ stetig folgt mit $j\to\infty$ + $$ \left| \frac{\lambda(Q_j)}{\lambda^n(Q_j)} - |\det{D\phi(x_0)}|\right| = \frac{1}{\lambda^n(Q_j)} \left| \int\limits_{Q_j} (|\det{D\phi(x)}|-|\det{D\phi(x_0)})dx\right| \to 0 \text{ mit } j\to \infty$$ + Zusammen mit Lemma VII.2 folgt: + $$ \liminf\limits_{j\to\infty} \frac{\lambda(Q_j)}{\mu(Q_j)} = \liminf\limits_{j\to\infty} \left( \frac{\lambda(Q_j)}{\lambda^n(Q_j)} \cdot \frac{\lambda^n(Q_j)}{\mu(Q_j)}\right) \geq 1 $$ + Widerspruch zu $(\ast\ast)$ \\ + $\implies \lambda\geq\mu$ auf $\script{A}$ + Jetzt betrachte $f: V\to \mathbb{\bar{R}}$. Es gilt $\{f \circ \phi < s \} = \phi^{-1}(\{f < s \}) \implies $ Mit $f$ ist auch $f\circ \phi$ $\lambda^n$-messbar und umgekehrt. \\ + Für $\lambda^n$-messbares $f: V\to [0,\infty]$ gilt: + \begin{align*} + (\ast\ast\ast) && \int\limits_U f(\phi(x)) |\det{D\phi(x)} dx \geq \int\limits_V f(y) dy + \end{align*} + Denn zuerst folgt $(\ast\ast\ast)$ aus $\lambda\geq\mu$ mit $f = \psi_B$, indem wir $A = \phi^{-1}(B)$ wählen \\ + $\rightarrow (\ast\ast\ast)$ gilt für nichtnegative $\lambda^n$-Treppenfunktionen und damit für alle $f\geq 0$ mit Satz IV.9 und Satz IV.10. + \item[Ziel:] Gleichheit in $(\ast\ast\ast)$ für $f\geq 0$ \\ + $\psi:V\to U$, $\psi = \phi^{-1}$ $C^1$-Diffeo \\ + $g = f\circ\phi |\det{D\phi}| $ \\ + Es gilt $1 = \det{D(\phi\circ\psi)}(y) = (\det{D\phi})(\psi(y)) \cdot \det{D\psi}(y)$ \\ + \begin{align*} + \implies \int\limits_V f(y)dy &= \int\limits_V g(\psi(y)) |\det{D\psi(y)} dy \\ + &\geq \int\limits_U g(x) dx \\ + &= \int\limits_U f(\phi(x)) |\det{D\phi(x)} dx + \end{align*} + $\implies \int\limits_V f(y)dy = \int\limits_U f(\phi(x)) |\det{D\phi(x)}| dx \rightarrow$ (2) für $f\geq 0$ \\ + Gleichung (1) folgt darauf mit $f = \psi_{phi(A)}$ \\ + Für $f$ beliebig zerlege $f = f^+ -f^-$ \end{proof} \sidenote{Vorlesung 20}{22.01.2021} @@ -113,9 +174,6 @@ \chapter{Der Transformationssatz} 0 & r^2 & 0 \\ 0 & 0 & r^2 \sin^2(\Theta) \end{array}\right)$$ - - \newpage - Allgemein gilt:\\ $g(x)$ ist symmetrisch und strikt positiv definit, denn $$ \ = \ \ = |D\Phi(x) v|^2 > 0$$ @@ -137,7 +195,23 @@ \chapter{Der Transformationssatz} \end{enumerate} \end{theorem} \begin{proof} - siehe Aufschrieb + $< (\triangledown v) \circ \phi, \frac{\partial \phi}{\partial x_i} > = (Dv) \circ \phi \cdot (D\phi \cdot e_i) = D(v\circ \phi) \cdot e_i = Du \cdot e_i = \frac{\partial u}{\partial x_i} \\ + \implies$ 1) folgt aus $y\circ \phi = D\phi \cdot x$, $y = \triangledown v$ \\ + Allgemein gilt für ein Vektorfeld $X: \script{U} \to \mathbb{R}^n$ \\ + $ = \sum\limits_{j=1}^n < D\phi \triangledown_gu, \frac{partial \phi}{\partial x_j} > x_j = \sum\limits_{j=1}^n \frac{\partial u}{\partial x_j} x_j$ \\ + \underline{2):} $\psi \in C^1_c(V)$ bel. $\script{C} = \psi\circ\phi$ \\ + \begin{align*} + \int\limits_V \psi \div{y}dy &= -\int\limits_V <\triangledown \psi, y > dy \\ + &= -\int\limits_{\script{U}} <(\triangledown \psi)\circ \phi, y\circ\phi > \sqrt{\det{g}} dx \\ + &= -\int\limits_{\script{U}} < D\phi \cdot \triangledown_g\script{C}, D\phi\cdot x > \sqrt{\det{g}} dx \\ + &= -\int\limits_{\script{U}} \sum\limits_{j=1}^n \frac{\partial \script{C}}{\partial x_j} x_j \sqrt{\det{g}} dx \\ + &= \int\limits_{\script{U}} \script{C} \frac{1}{\det{g}} \frac{\partial}{\partial x_j} (\sqrt{\det{g}}x_j) \sqrt{\det{g}} dx \\ + &= \int\limits_{\script{U}} \script{C} div_g x \sqrt{\det{g}} dx \\ + &= \int\limits_V \psi div_g x\circ \phi^{-1} dy + \end{align*} + Übung Fundamentallemma Varationsrechnung \\ + $\implies div{y\circ\phi} = div_g x \implies$ 2) \\ + 3) folgt aus Kombination von 1) und 2) mit $g^{ij} = g^{ji}$ \end{proof} \begin{example}[Laplace in Polarkoordinaten im $\mathbb{R}^3$] diff --git a/chapter8.tex b/chapter8.tex index abd2728..7cd4b51 100644 --- a/chapter8.tex +++ b/chapter8.tex @@ -60,7 +60,12 @@ \chapter{Das Flächenmaß auf Untermannigfaltigkeiten} $$A(f, \Phi(E)) = A(f\circ\Phi, E) \ \ \ \forall \ E \subseteq \script{U} \ \lambda^n\text{-messbar}$$ \end{theorem} \begin{proof} - siehe Aufschrieb + Kettenregel \\ + $D(f\circ \phi)(x)^T D(f\circ\phi)(x) = D\phi(x)^T Df(\phi(x))^T Df(\phi(x)) D\phi(x)$ \\ + Determinanten davon ist \\ + $\det{(D(f\circ\phi)(x)^T D(f\circ\phi)(x))} = |\det{(D\phi(x))}|^2 \det{(Df(\phi(x))^T Df(\phi(x)))} \\ + \implies J(f\circ\phi)(x) = J f(\phi(x)) |\det{(D\phi(x))}|\\ + \implies $ Beh. folgt aus Trafo-Formel \end{proof} \begin{definition}[Untermannigfaltigkeiten] @@ -87,8 +92,18 @@ \chapter{Das Flächenmaß auf Untermannigfaltigkeiten} \begin{theorem} Jede $n$-dimensionale Untermannigfaltigkeit $M \subseteq \mathbb{R}^{n+k}$ ist als abzählbare Vereinigung $M = \bigcup\limits_{i\in\mathbb{N}} K_i$ von kompakten Mengen darstellbar. \end{theorem} + \begin{proof} - siehe Aufschrieb + \item[I] $M\cap \overline{B_r(p)}$ ist kompakt für $p\in M$ und $r>0$ hinreichend klein. Sei dazu $\phi: W\to \phi(W)$ lokalte Plättung mit $p\in W$ \\ + $\phi: W\to \mathbb{R}^{n+k}$ ist stetig $\implies M\cap W = \phi^{-1}(\mathbb{R}^n\times\{0\})$ ist abgeschlossen in $W$. Für $\overline{B_r(p)}\subset W$ ist damit $M\cap \overline{B_r(p)}$ abgeschlossen, also kompakt. Setze nun für $p\in M$: + $$ r(p) = \sup\{r >0: M\cap \overline{B_r(p)} \text{ ist kompakt } \} \in (0,\infty] $$ + $\implies M\cap \overline{B_r(p)}$ ist kompakt $\forall r\in r(p)$ \\ + (Abg. Teilmengen einer kompakten Menge sind kompakt) + \item[II] $r(p) \leq \liminf\limits_{i\to\infty}r(p_i)$ für $p,p_i\in M$ mit $p_i \to p$. \\ + Zu $r < r(p)$ wähle $R\in (r, r(p))$ \\ + Es gilt: $M\cap \overline{B_r(p)} \subset M\cap \overline{B_R(p)}$ für $i$ groß $\implies r(p_i) \geq r(p)$. \\ + Für $P\subset M$ dicht ist damit $M$ Vereinigung der kompakten Teilmengen \\ $M = \bigcup\limits_{p\in P} M\cap \overline{B_{\frac{r(p)}{2}}(p)}$ \\ + Betrachte nun $Q_{j,l} = 2^{-l} (j+[0,1]^{n+k})$ für $j\in \mathbb{Z}^{n+k}$, $l\in\mathbb{N}_0$. Wähle in jedem Würfel $Q_{j,l}$, der $M$ trifft, einen Punkt $p_{j,l}\in M$. Die Menge $P$ dieser $p_{j,l}$ ist abzählbar und dicht in $M$. \end{proof} \begin{definition} @@ -100,7 +115,10 @@ \chapter{Das Flächenmaß auf Untermannigfaltigkeiten} $C^1$-Parametrisierungen $f_i:\script{U}_i \to M$, wobei $i\in\mathbb{N}$, sodass $M = \bigcup\limits_{i \in \mathbb{N}} f_i(\script{U}_i)$ \end{lemma} \begin{proof} - siehe Aufschrieb + $\forall p\in M$ $\exists$ lokale Plättung $\phi: W \to \phi(W)$ mit $p\in W$ \\ + $\implies \script{U} = \mathbb{R}^n \cap \phi(W)$ ist offen in $\mathbb{R}^n$ und $f = \phi^{-1}|_{\mathbb{R}\cap \phi(W)}$ ist eine lokalte Parametrisierung von $M$ mit $p\in f(\script{U})$ \\ + Außerdem ist $f(\script{U}) = M\cap W$ offen in $M$. \\ + $K\subset M$ kompakt wird also durch endlich viele Bild(?) überdeckt. Beh. folgt aus Satz VIII.6 \end{proof} \begin{theorem} @@ -111,7 +129,15 @@ \chapter{Das Flächenmaß auf Untermannigfaltigkeiten} \end{enumerate} \end{theorem} \begin{proof} - siehe Aufschrieb + \item[1)] s. Blatt 11 (Satz VIII.5) \\ + \item[2)] O.E. $V_1 \cap V_2 \subset W$ für eine Plättung $\phi: W \to \phi(W)$ denn $C^1$-Eigenschaft ist lokal. \\ + Betrachte $\phi \circ f_i: f_i^{-1}(V_1\cap V_2) \to \phi(V_1\cap V_2)$ $i=1,2$ \\ + Es gilt: $f_2^{-1} \circ f_1 = (\phi\circ f_2)^{-1} \circ (\phi\circ f_1)$ auf $f_1^{-1}(V_1\cap V_2)$ \\ + \underline{z.z} $\phi\circ f_i$ ist Diffeo \\ + $\phi\circ f_i$ ist definiert, injektiv und es gilt $Rang (D(\phi\circ f_i)) (x) = Rang (D\phi(f_i(x)) Df(x)) = n$ $\forall x\in f_i^{-1}(V_1 \cap V_2) \\ + \overset{\text{Satz von der lokalten Umkehrbarkeit}}{\implies} \phi(V_1 \cap V_2)$ ist offen und $\phi\circ f_i: f_i^{-1}(V_1\cap V_2) \to \phi(V_1 \cap V_2)$ ist Diffeo. \\ + ($\phi(V_1\cap V_2)\subset \mathbb{R}^n\times \{0\} \subset \mathbb{R}^{n+k}$) + \end{proof} \begin{theorem}[Flächenmaß] @@ -121,14 +147,41 @@ \chapter{Das Flächenmaß auf Untermannigfaltigkeiten} $$\mu_M(E) = \int\limits_{f^{-1}(E)}Jf(x) \ dx$$ \end{theorem} \begin{proof} - siehe Aufschrieb - \end{proof} + \item[Trivial:] $\varnothing \in M$ \\ + Für jede lok. Para. $f\script{U} \to M$ gilt: $f^{-1}(M\setminus E) = \script{U}\setminus f^{-1}(E)$ und $f^{-1}(\bigcup\limits_{i=1}^\infty E_i) = \bigcup\limits_{i=1}^\infty f^{-1}(E_i) \\ + \implies M$ ist $\sigma$-Algebra. \\ + Mit $V$ offen in $M$ ist $f^{-1}(V)$ offen im $\mathbb{R}^n \implies M$ enthält alle Borelmengen. \\ + Sei nun $M = \bigcup\limits_{i=1}^\infty M_i$ disjunkte Zerlegung in Mengen $M_i\in M$, sodass $M_i \subset V_i$ für lok. Para. $f_i: \script{U_i}\to f(\script{U_i}) = V_i$ \\ + Solch eine Zerlegung ex., denn nach lemma VIII.8 $\exists$ lok. para $f_i:\script{U_i} \to f(\script{U_i}) = V_i$ mit $M = \bigcup\limits_{i=1}^\infty V_i$ und wir wählen jetzt die Borelmengen $M_i = V_i \setminus \bigcup\limits_{j=1}^{i-1}V_j$. \\ + Aus den Eigenschaften folgt für das gesuchte Maß $\mu_M$ und alle $E\in M$ + \begin{align*} + (\ast) && \mu_M(E) = \sum\limits_{i=1}^\infty \mu_M(E\cap M_i) = \sum\limits_{i=1}^\infty \int\limits_{f_i^{-1}(E\cap M_i)} J f_i(x) dx + \end{align*} + $\implies$ Eindeutigkeit von $\mu_M$. \\ + Durch $(\ast)$ wird ein Maß auf $M$ definiert. \\ + Ist $E = \bigcup\limits_{j=1}^\infty E_j$ mit $E_j$ messbar und paarweise disjunkt so folgt + \begin{align*} + \mu_M(E) &= \sum\limits_{i=1}^\infty \int\limits_{f_i^{-1}(M_i\cap E)}Jf_i(x) dx \\ + &= \sum\limits_{i,j=1}^\infty \int\limits_{f_i^{-1}(E_j\cap M_i)} Jf_i(x)dx \\ + &= \sum\limits_{j=1}^\infty \mu_M(E_j) + \end{align*} + Ist $f:\script{U} \to V = f(\script{U})$ beliebige lokale Para und $E\subset V$ messbar, so ist $\phi_i = f_i^{-1}\circ f: f^{-1}(V\cap V_i) \to f_i^{-1}(V\cap V_i)$ ein $C^1$-Diffeo zw. offenen Mengen nach Satz VIII.9(2). Aus Satz VIII.3 folgt \begin{align*} + \mu_M(E) &= \sum\limits_{i=1}^\infty \int\limits_{f_i^{-1}(E\cap M_i)} Jf_i(x) dx \\ + &= \sum\limits_{i=1}^\infty \int\limits_{\phi_i\circ f^{-1}(E\cap M_i)} Jf_i(x) dx \\ + &= \sum\limits_{i=1}^\infty \int\limits_{f^{-1}(E\cap M_i)}J(f_i\circ\phi_i)(x) dx \\ + &= \int\limits_{f^{-1}(E)} Jf(x) dx + \end{align*} - \newpage + \end{proof} + \newpage \begin{theorem}[Oberflächenintegral] Sei $M \subseteq \mathbb{R}^{n+k} n$-dimensionale $C^1$-Untermannigfaltigkeit und $M = \bigcup\limits_{i \in \mathbb{N}} M_i$ eine paarweiße disjunkte, messbare Zerlegung mit $M_i \subseteq \script{V}_i$ für lokale Parametrisierungen $f_i: \script{U}_i \to \script{V}_i$. Für eine messbare Funktion $u: M \to \bar{\mathbb{R}}$ gilt: $$\int\limits_M u \ d\mu_M = \sum\limits_{i \in \mathbb{N}} \int\limits_{f_i^{-1}(M_i)} u(f_i(x))\ Jf_i(x)\ dx$$ \end{theorem} + \begin{proof} + Aussage gilt nach Satz VIII.10 für $u$ = Charakteristische Funktion\\ $\implies$ gilt auch für messbare Treppenfunktionen. \\ + Für $u\geq 0$ folgt Beh. durch Approximation von unten durch Treppenfunktionen (Satz Mon. Konv.) [Auf der rechten Seite benutze zuerst Mon. Konv. für die einzelnen Integrale und dann für die Reihe] \\ Für $u$ integrierbar zerlege $u = u^+ - u^-$ + \end{proof} \sidenote{Vorlesung 23}{01.02.2021} \begin{lemma} @@ -141,7 +194,13 @@ \chapter{Das Flächenmaß auf Untermannigfaltigkeiten} \end{enumerate} \end{lemma} \begin{proof} - siehe Aufschrieb + Zu $q\in N$ wähle lokalte Plättung $\phi: W \to \phi(W)$ von $M$ mit $T^{-1}(q) \in W$. Dann ist $\phi \circ T^{-1}: T(W) \to \phi(W)$ eine lokalte Plättung von $N$ $\implies N$ ist $C^1$-Umge. \\ + Ist $f: \script{U} \to M$ lokale Parametrisierung, so auch $T\circ f: \script{U}\to W$ und umgekehrt. Wegen $(T\circ f)^{-1} (T(A)) = f^{-1}(A)$ ist $A\subset M$ $\mu_M$-messbar $\Leftrightarrow T(A) \subset N$ $\mu_M$-messbar. + \item[1)] O.E. $A\subset f(\script{U})$ (s. Beweis zu Satz VIII.10) für $f: \script{U}\to M$ lokalte Parametriersung. $DT(x) = \lambda Q$ \\ + $\implies D(T\circ f)(x)^T D(T\circ f)(x) = Df(x)^T DT(f(x))^T DT(f(x))Df(x) = \lambda^n Df(x)^T Df(x)$ \\ + $\overset{\text{Satz VIII.11}}{\implies} \mu_M (T(A)) = \int\limits_{(T\circ f)^{-1}(T(A))} J(T\circ f)(x) dx = \lambda^n\int\limits_{f^{-1}(A)}Jf(x) dx = \lambda^n \mu_M(A)$ + \item[2)] Für $u = \psi_B$ mit $B\subset N$ $\mu_N$-messbar folgt 2) aus 1). + Durch Approximation mit Treppenfunktionen von unten folgt 2) für $u\geq 0$ und für $u$ belibig zerlege $u = u^+ - u^-$ \end{proof} \begin{theorem}[Zwiebelformel] @@ -153,10 +212,19 @@ \chapter{Das Flächenmaß auf Untermannigfaltigkeiten} \end{align*} \end{theorem} \begin{proof} - siehe Aufschrieb + Sei $f: \script{U}\to V\subset S^n$ lok. Para. und $C(V) = \{rw: w\in V, r> 0 \}$ sei der offene Kegel über $V$. \\ + Betrachte Diffeo $\phi: (0,\infty) \times \script{U} \to C(V)$, $\phi(v,x) = rf(x)$. \\ + Mit $g_{i,j}(x) = < \frac{\partial f(x)}{\partial x_i}, \frac{\partial f(x)}{\partial x_j}>$ gilt $D\phi (r,x)^T D\phi (r,x) = \begin{pmatrix} 1 & 0 \\ 0 & r^2 g(x) \end{pmatrix} \in \mathbb{R}^{n+1}\times \mathbb{R}^{n+1}$ \\ + Ist $E = f(A)$ für $\lambda^n$-messbares $A\subset \script{U}$ und $C(E)$ der Kegel über $E$, so folgt + \begin{align*} + \int\limits_{C(E)} u(p) d\lambda^{n+1}(p) &= \int\limits_{(0,\infty)\times A} u(rf(x)) r^n \sqrt{det{(g(x))}} d\lambda^{n+1}(r,x) \\ + &= \int\limits_0^\infty r^n \int\limits_A u(rf(x)) \sqrt{\det{(g(x))}} d\lambda^n(x) d\lambda^1(r) \\ + &= \int\limits_0^\infty r^n \int\limits_E u(rw) d\mu_{S^n}(w) d\lambda^1(r) \\ + &= \int\limits_0^\infty \int\limits_{\{rw: w\in E\} } u(p) d\mu_{\partial B_r(0)}(p) d\lambda^1(r) + \end{align*} + Wähle nun disjunkte Zerlegung $S^n = \bigcup\limits_{j=1}^N E_j$ mit $E_j \subset V_j$, wobei $f_j: \script{U_j}\to V_j$ lok. Para. ist. Durch Addition folgt Beh. \end{proof} - \newpage \begin{example} Mit $u = \psi_{B_1(0)}$ folgt für $w_n = \mu_{S^n}(S^n)$: $$\alpha_{n+1} = \lambda^{n+1}(B_1(0)) = \int\limits_0^1 \mu_{\partial B_r(0)} (\partial B_r(0)) \ dr = \int\limits_0^1 w_n r^n \ dr = \frac{w_n}{n+1}$$ diff --git a/chapter9.tex b/chapter9.tex index c38b4ef..b86e5cf 100644 --- a/chapter9.tex +++ b/chapter9.tex @@ -9,7 +9,7 @@ \chapter{Der Integralsatz von Gauß} $\forall \ p \in \partial \Omega \ \exists \ W \subseteq \mathbb{R}^n$ offen mit $p \in W$ und $\Phi: W\to \Phi(W)$\\ $C^1$-Diffeomorphismus mit: $$\Phi(W \cap \Omega) = \mathbb{H}^n \cap \Phi(W) \text{ wobei } \mathbb{H}^n = \mathbb{R}^{n-1} \times (-\infty, 0)$$ - \includegraphics[width=.8\textwidth]{img/IX_1_Plättung.png} + \includegraphics[width=.8\textwidth]{img/IX_1_Plaettung.png} \item Subniveau:\\ $\forall p\in \partial \Omega \ \exists W \subset \mathbb{R}^n$ offen mit $p \in W$ und $h \in C^1(W)$ mit $Dh(q) \neq 0) \\\forall q \in W$, sodass $\Omega \cap W = \{q \in W \ | \ h(q) < 0\}$ \item Subgraph:\\ @@ -19,10 +19,9 @@ \chapter{Der Integralsatz von Gauß} Die Menge $\Omega$ hat $C^1$-Rand wenn eines (und damit jedes) der drei Kriterien erfüllt ist. \end{theorem} \begin{proof} - siehe Aufschrieb + siehe Blatt 12 \end{proof} -\newpage \begin{lemma} In der Situation von Satz IX.1,3) gilt: \begin{align*} @@ -32,7 +31,12 @@ \chapter{Der Integralsatz von Gauß} $\implies \partial \Omega$ ist $(n-1)$-dimensionale $C^1$-Untermannigfaltigkeit des $\mathbb{R}^n$ nach dem Graphenkriterium bei Untermannigfaltigkeiten. \end{lemma} \begin{proof} - siehe Aufschrieb + Mit $h: \script{U} \times I \to \mathbb{R}$, $h(x,y) = y-u(x)$ \\ + Vor. $\implies \Omega \cap (\script{U}\times I) = \{ h<0 \}$ \\ + $h$ stetig $\implies \partial \Omega \cap (\script{U}\times I) \subset \{ h=0\}$ \\ + Ist $h(x,y) = 0$ so folgt für $\epsilon$ klein $(x,y-\epsilon)\in\script{U}\times I$ und $h(x,y-\epsilon) = h(x,y) - \epsilon < 0 \rightarrow (x,y-\epsilon)\in\Omega \cap (\script{U}\times I)\\ + \implies (x,y) \in \bar{\Omega}$ und wegen $(x,y) \notin \Omega \implies (x,y) \in \partial\Omega \cap (\script{U}\times I) \implies \partial\Omega \cap (\script{U}\times I) = \{ h=0\}$ \\ + Zweite Aussage folgt aus $\bar{\Omega} = \Omega \cup \partial\Omega$ \end{proof} \sidenote{Vorlesung 24}{05.02.2021} @@ -50,6 +54,18 @@ \chapter{Der Integralsatz von Gauß} $T_pM = \{$Alle Tangentualvektoren$\}$ $(n-1)$-dimensionaler Untervektorraum. \end{remark} +\begin{proof} + Wähle mit Satz IX.1,3) nach ? Umnummerierung der Koordinaten $\Omega \cap (\script{U}\times I) = \{ (x,y)\in \script{U}\times I: y < u(x) \}$ \\ + Sei $q\in \partial\Omega \cap (\script{U}\times I)$ \\ + Lemma IX.2 $\implies f:\script{U} \to \mathbb{R}^n$, $f(x) = (x, u(x))$ ist lokale Para. von $\partial\Omega$ \\ + $T_q(\partial\Omega)$ in $q = (x, u(x))$ hat Basis $(e_i, \frac{\partial u}{\partial x_i}(x)) = \frac{d}{dt} f(x+te_i)|_{t=0}$ für $i=1, ..,n-1$\\ + Definiere auf $\partial\Omega \cap (\script{U}\times I)$ für $q=(x,u(x))$ $$r(q) = \frac{(-Du(x), 1)}{\sqrt{(1+||Du(x)||^2)}}$$ + $\implies r$ erfüllt 1) \\ + Es gilt $q+t r(q) = (x(t),y(t))$ mit $x(t) = x-t\frac{Du(x)}{\sqrt{(1+||Du(x)||^2)}}$, $y(t) = u(x) + t \frac{1}{\sqrt{(1+||Du(x)||^2)}}$ \\ + $\implies \frac{d}{dt} |y(t)-u(x(t))|_{t=0} = \frac{1}{\sqrt{(1+||Du(x)||^2)}} + Du(x) \frac{Du(x)}{\sqrt{(1+||Du(x)||^2)}} = \sqrt{(1+||Du(x)||^2)} > 0$ \\ + Lemma IX.2 $\implies q+t r(q) \notin\Omega$, $q-tr(q)\in\Omega$ für $t$ hinreichend klein \\ + $\implies r\in C^0$ folgt aus $u\in L^1$ +\end{proof} \begin{definition} Für $\Omega \subseteq \mathbb{R}^n$ sei $C^1(\bar{\Omega})$ der Unterraum aller $f\in C^1(\Omega)$, für die $f$ und $Df$ stetige Fortsetzungen auf $\partial \Omega$ besitzen. Die \textbf{Fortsetzung} wird wieder mit $f$ bezeichnet. \end{definition} @@ -59,7 +75,24 @@ \chapter{Der Integralsatz von Gauß} $$\int\limits_{\Omega} div \ X \ d\lambda^n = \int\limits_{\partial \Omega} \ d\mu$$ \end{lemma} \begin{proof} - siehe Aufschrieb + $X(x,a) = 0$ $\forall x\in \script{U} \implies \int\limits_\Omega \frac{\partial X_n}{dy} \overset{\text{Fubini}}{=} \int\limits_{\script{U}}\int\limits_u^{u(x)} \frac{\partial X_n}{\partial y} dy dx = \int\limits_{\script{U}} X_n(x,u(x)) dx$ \\ + \item[\underline{Beh}] Für $i=1,...,n-1$ gilt: $\int\limits_\Omega \frac{\partial X_i}{\partial x_i} d\lambda^n = - \int\limits_{\script{U}} X_i(x,u(x))\frac{\partial u}{\partial x_i}dx$ \\ + Damit gilt dann: + \begin{align*} + \int\limits_\Omega div X d\lambda^n &= -\sum\limits_{i=1}^{n-1} \int \limits_{\script{U}} X_i(x,u(x)) \frac{\partial u}{\partial x_i}(x) dx + \int\limits_{\script{U}} X_n(x,u(x)) dx \\ + &= \int\limits_{\script{U}} \sqrt{(1+||Du(x)||^2)}dx \\ + &= \int\limits_{\partial\Omega} d\mu + \end{align*} + \item[] \underline{Beweis von Beh.} \\ + Sei $\eta \in C^\infty(\mathbb{R})$ mit $\eta(t) = 1$ für $t \leq -2$, $eta(t) = 0$ für $t\geq -1$ und setze $eta_\epsilon (t):= \eta(\frac{t}{\epsilon})$ \\ + $\implies \eta_\epsilon(y-u(x)) = 0$ für $y \geq u(x) - \epsilon$ und $\lim\limits_{\epsilon\to 0} \eta_\epsilon(y-u(x)) = \begin{cases} 1 \text{ falls } y < u(x) \\ 0 \text{ sonst} \end{cases}$ \\ + $\int\limits_\Omega \frac{\partial X_i}{\partial x_i} (x,y) \eta_\epsilon(y-u(x)) d\lambda^n(x,y) = \int\limits_\Omega X_i(x,y) \eta'_\epsilon(y-u(x)) \frac{\partial u}{\partial x_i}(x)d\lambda^n(x,y) = -\int\limits_\Omega \frac{\partial X_i}{\partial y}(x,y) \eta_\epsilon(y-u(x)) \frac{\partial u}{\partial x_i}x_i d\lambda^n(x,y) $ \\ + Alle Funktionen beschränkt $\overset{\text{Lebesgue}, \epsilon\to 0}{\implies}$ + \begin{align*} + \int\limits_\Omega \frac{\partial X_i}{\partial x_i} d\lambda^n &= -\int\limits_\Omega \frac{\partial X_i}{\partial y} (x,y) \frac{\partial u}{\partial x_i} (x) d\lambda^n(x,y) \\ + &= -\int\limits_{\script{U}} \left( \int\limits_u^{u(x)} \frac{\partial X_i}{\partial y} (x,y) dy\right) \frac{\partial u}{\partial x_i} dx \\ + &= -\int\limits_{\script{U}} X_i (x,u(x)) \frac{\partial u}{\partial x_i} dx + \end{align*} \end{proof} \begin{lemma} @@ -70,16 +103,18 @@ \chapter{Der Integralsatz von Gauß} \end{enumerate} \end{lemma} \begin{proof} - siehe Aufschrieb + Wähle eine beschränkte offene Menge $\Omega \supset K$ und bestimme zu jedem $p\in \bar{\Omega}$ einen Ball $b_{r(p)}(p)$ wie folgt: \\ + Für $p\in K$ wähle $\lambda(p)\in \Lambda$ mit $p\in W_{\lambda(p)}$ und weiter $r(p) > 0$ mit $\overline{B_{2r(p)}(p)}\subset (W_{\lambda(p)}\cap \Omega)$. \\ + Für $p\in \bar{\Omega}\setminus K$ wähle $r(p) >0 $ mit $\overline{B_{2r(p)}(p)}\cap K = \varnothing$ endlich viele Kugeln $B_{r(p_j)}$, $1\leq j \leq N$, Überdeckungen $\bar{\Omega}$. Wähle nun $\tilde{\psi_j} \in C^\infty_C(\mathbb{R}^n)$ mit $\tilde{\psi_j} = \begin{cases} 1 \text{ auf } B_{r(p_j)}(p_j) \\ 0 \text{ auf } \mathbb{R}^n\setminus B_{2r(p_j)}(p_j) \end{cases} \\ + \implies \sum\limits_{j=1}^N \tilde{\psi_j} \geq 1$ auf $\bar{\Omega}$. \\ + Die Funktionen $\psi_j = \frac{\tilde{\psi_j}}{\sum\limits_{j=1}^N \tilde{\psi_j}}$ mit $j\in J:=\{j: p_j \in K \}$ sind glatt in $\Omega$ mit $spt \psi_j \subset\subset W_{\lambda(p_k)} \cap \Omega $ wegen $\tilde{\psi_j}|_K = 0$ für $j\notin J$ \\ + $\implies \sum\limits_{j\in J} \psi_j = 1$ auf $K$. \end{proof} \begin{theorem}[Integralsatz von Gauß] $\Omega \subseteq \mathbb{R}^n$ offen, beschränkt mit $C^1$-Rand und äußere Normale $\nu:\partial\Omega\to\mathbb{R}^n$. Dann gilt für $X\in C^1(\bar{\Omega}, \mathbb{R}^n)$: $$\int\limits_{\Omega} div \ X \ d\lambda^n = \int\limits_{\partial\Omega} \ d\mu$$ \end{theorem} -\begin{proof} - siehe Aufschrieb -\end{proof} \begin{remark} Gilt auch für Gebiete, deren Rand lokal lipschitz ist (d.h. $u$ in Satz IX.1,3) ist lipschitz (siehe Buch von H.W. Alt: Lineare Funktionalanalysis) \\ @@ -87,13 +122,19 @@ \chapter{Der Integralsatz von Gauß} \end{remark} \sidenote{Vorlesung 25}{08.02.2021} +\begin{proof} + Wähle nach Satz IX.1 3) zu jedem $p\in \partial\Omega$ eine Umgebung $W_p$, in der $\Omega$ bzgl geeigneter Koordinaten als Subgraph dargestellt ist. Für $p\in \Omega$ setze einfach $W_p = \Omega \implies \bigcup\limits_{p\in\bar{\Omega}} W_p \supset \bar{\Omega}$ und $W_p$ sind offen $\forall p\in \bar{\Omega}$ \\ + Lemma IX.6 $\implies\exists$ untegeordnete Zerlegung der 1 $\psi_1, ..., \psi_N \in C^\infty_C (\mathbb{R}^n)$. \\ + Liegt $spt \psi_j$ in einer Randumgebung $W_p = \script{U}\times I$ wie in Satz IX.1 3), so folgt aus Lemma IX.5 $$ \int\limits_\Omega div (\psi_j X) dx = \int\limits_{\partial\Omega} < \psi_j X, v> d\mu$$ + Ist $spt \psi_j \in \Omega$, so folgt mit part. Integration $$\int\limits_\Omega div(\psi_j X) dx = 0 = \int\limits_{\partial\Omega} <\psi_j X, v> d\mu$$ + $\implies \int\limits_\Omega div X dx = \sum\limits_{j=1}^N \int\limits_\Omega div(\psi_j X) dx = \sum\limits_{j=1}^N \int\limits_{\partial\Omega} <\psi_j X, v> d\mu = \int\limits_{\partial\Omega} d\mu$ +\end{proof} \begin{example} Wähle $X(x) = x \implies \lambda^n(\Omega) = \frac{1}{n} \int\limits_{\Omega} div \ X \ dx = \frac{1}{n} \int\limits_{\partial \Omega} \ d\mu(x)$\\ Speziell: $\alpha = \frac{\omega_{n-1}}{n}$\\ $\Omega = B_1(0) \subseteq \mathbb{R}^n \rightarrow \nu(x) = x, \partial\Omega = S^{n-1}$ \end{example} -\newpage \begin{lemma}[Greensche Formeln] $\Omega \subseteq \mathbb{R}^n$ offen und beschränkt mit $C^1$-Rand. Dann gilt für $u \in C^1(\bar{\Omega})$ und $v \in C^2(\bar{\Omega})$ \begin{enumerate} @@ -106,7 +147,10 @@ \chapter{Der Integralsatz von Gauß} \end{lemma} \begin{proof} - siehe Aufschrieb + \item[1)] Benutze $div(u\triangledown v) = <\triangledown u, \triangledown v> + u \triangle v$ \\ + Gauss $\implies \int\limits_{\partial\Omega} d\mu = \int\limits_\Omega div(u \triangledown v) d\lambda^n = \int\limits_\Omega (<\triangledown u, \triangledown v> + u \triangle v) d\lambda^n$ + \item[2)] 1) $\implies \int (v\triangle u + <\triangledown v, \triangledown u>) d\lambda^n = \int\limits_{\partial\Omega} v \frac{\partial u}{\partial v} d\mu$ \\ + Bilde Differenz $\implies$ 2) \end{proof} \begin{example} diff --git "a/img/IX_1_Pl\303\244ttung.png" b/img/IX_1_Plaettung.png similarity index 100% rename from "img/IX_1_Pl\303\244ttung.png" rename to img/IX_1_Plaettung.png