From 2a6c90cd6a75f2bea04f6f47205334bcd0c1b703 Mon Sep 17 00:00:00 2001 From: christian Date: Fri, 6 Nov 2020 17:52:10 +0100 Subject: [PATCH] add ana --- ws2019/ipi/uebungen/virtuals.cpp | 14 ++++- ws2020/ana/uebungen/ana1.pdf | Bin 0 -> 95553 bytes ws2020/ana/uebungen/ana1.tex | 95 +++++++++++++++++++++++++++++++ 3 files changed, 108 insertions(+), 1 deletion(-) create mode 100644 ws2020/ana/uebungen/ana1.pdf create mode 100644 ws2020/ana/uebungen/ana1.tex diff --git a/ws2019/ipi/uebungen/virtuals.cpp b/ws2019/ipi/uebungen/virtuals.cpp index 31d96c6..3ffb8eb 100644 --- a/ws2019/ipi/uebungen/virtuals.cpp +++ b/ws2019/ipi/uebungen/virtuals.cpp @@ -11,13 +11,24 @@ class A { class B : public A { public: + using A::a; void b() { printf("B::b()\n");}; virtual void vb() {printf("B::vb()\n"); }; void a(double d) {printf("B::a(double %f)\n", d); }; void a(int i) {printf("B::a(int %d)\n", i); }; virtual void va() {printf("B::va()\n"); }; + B operator+(int n); + friend B operator+(int n, B& b); }; +B B::operator+(int n) { + return *this; +} + +B operator+(int n, B& b) { + return b + n; +} + class C : public B { public: virtual void c() {printf("C::c()\n"); }; @@ -29,6 +40,7 @@ class C : public B { int main() { // A a; A abstrakte Klasse B b; + B b2; C c; A* pa=&b; B* pb=&c; @@ -40,7 +52,7 @@ int main() { // pa->b(); pa ist vom Typ A* und kennt deswegen b() nicht // pa->vb(); pa ist vom Typ A* und kennt deswegen vb() nicht pa->a(x); // B::a(int), hier wird double x implizit zu int gecastet - // pb->a(); pb ist vom Typ B*, hier ist A::a() überladen durch + pb->a(); //pb ist vom Typ B*, hier ist A::a() überladen durch // entweder B::a(double) bzw. 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zN9T7aQ`44`Zl<;z@F&uVkxd zwO_ZXo$9pcX)osurDb#L&}L?HD;vD7GSsm;-Xr_+Sq|_2{x(;%g$vrl<>q@qgdqYD M;-^m)UMLd(7m3@F1^@s6 literal 0 HcmV?d00001 diff --git a/ws2020/ana/uebungen/ana1.tex b/ws2020/ana/uebungen/ana1.tex new file mode 100644 index 0000000..2e0415c --- /dev/null +++ b/ws2020/ana/uebungen/ana1.tex @@ -0,0 +1,95 @@ +\documentclass[uebung]{../../../lecture} + +\begin{document} + +\punkte + +\title{Analysis III: Übungsblatt 1} +\author{Leon Burgard, Christian Merten} + +\begin{aufgabe}[] + Beh.: $\mathcal{A}_{\mu}$ ist eine $\sigma$-Algebra auf $X$. + \begin{proof} + \begin{enumerate}[(i)] + \item $X \in \mathcal{A}_{\mu}$, denn $\mu(\emptyset) = 0$, da + $\mu$ Maß und $X \triangle X = \emptyset$. Mit (ii) ist + auch $\emptyset = X^{C} \in \mathcal{A}_{\mu}$. + \item Sei $A \in \mathcal{A}_{\mu}$. Dann ex. + $B, C \in \mathcal{A}$ mit $\mu(C) = 0$ und + $A \triangle B \subset C$. Dann ist + \[ + A^{c} \triangle \underbrace{B^{c}}_{\in \mathcal{A}} + = A^{c} \setminus B^{c} \cup B^{c} \setminus A^{c} + = B \setminus A \cup A \setminus B + = A \triangle B + \subset C + .\] + Also $A^{c} \in \mathcal{A}_{\mu}$. + \item Sei $A_i \in \mathcal{A}_{\mu}$ für $i \in \N$. Dann + ex. $\forall i \in \N$ ein $B_i \in \mathcal{A}$ und + $\mu$-Nullmenge $C_i \in \mathcal{A}$, s.d. + $A_i \triangle B_i \subset C_i$. + + Betrachte nun $B := \bigcup_{i \in \N} B_i$ + und $C := \bigcup_{i \in \N} C_i$. Es ist + $B, C \in \mathcal{A}$ + da $\mathcal{A}$ $\sigma$-Algebra. Außerdem + ist $\mu(C) = 0$, wg. $\sigma$-Additivität von $\mu$. + Damit folgt + \begin{align*} + A \triangle B &= + \left( \bigcup_{i \in \N} A_i \right) + \triangle + \left( \bigcup_{i \in \N} B_i \right) \\ + &= + \left( \bigcup_{i \in \N} A_i \setminus \bigcup_{i \in \N} B_i \right) \cup \left( \bigcup_{i \in \N} B_i \setminus \bigcup_{i \in \N} A_i \right) \\ + &\subset + \left( \bigcup_{i \in \N} A_i \setminus B_i \right) \cup \left( \bigcup_{i \in \N} B_i \setminus A_i \right) \\ + &= \bigcup_{i \in \N} \left( A_i \setminus B_i \cup B_i \setminus A_i \right) \\ + &\subset C_i \\ + &= C + .\end{align*} + \end{enumerate} + \end{proof} + + Beh.: $\overline{\mu}$ ist ein Maß. + \begin{proof} + \begin{enumerate}[(i)] + \item Z.z.: $\overline{\mu}$ wohldefiniert. Sei dazu + $A \in \mathcal{A}_{\mu}$ und $B, B', C, C' \in \mathcal{A}$, + s.d. $A \triangle B \subset C$ und $A \triangle B' \subset C'$. + Definiere $\tilde{C} := C \cup C' \in \mathcal{A}$ + Es folgt direkt $\mu(\tilde{C}) = 0$. Es + gilt weiter + \begin{align*} + \tilde{C} \supset \underbrace{(A \triangle B)}_{\subset C} \triangle + \underbrace{(A \triangle B')}_{\subset C'} + &= \underbrace{A \triangle A}_{= \emptyset} \triangle B \triangle B' \\ + &= B \triangle B' \\ + &= B \setminus B' \cup B' \setminus B + .\end{align*} + Also insbesondere $B \setminus B' \subset \tilde{C}$ und + $B' \setminus B \subset \tilde{C}$. + Mit $B = (B \setminus B') \cup (B \cap B')$ disjunkt und + der $\sigma$-Additivität von $\mu$ folgt + \begin{align*} + \mu(B) &= \mu(\underbrace{B \setminus B'}_{\subset \tilde{C}}) + \mu(B \cap B') \\ + &= 0 + \mu(B \cap B') + 0 \\ + &= \mu(B \cap B') + \mu(\underbrace{B' \setminus B}_{\subset \tilde{C}}) \\ + &= \mu(B') + .\end{align*} + Also $\overline{\mu}$ wohldefiniert. + \item Z.z.: $\overline{\mu}(\emptyset) = 0$. Es ist + $\emptyset \triangle \emptyset = \emptyset$, also + $\overline{\mu}(\emptyset) = \mu(\emptyset) = 0$. + \item Die $\sigma$-Additivität folgt direkt aus der + $\sigma$-Additivität von $\mu$. + \end{enumerate} + \end{proof} +\end{aufgabe} + +\begin{aufgabe} + +\end{aufgabe} + +\end{document}