%AUTOR:Lucht CONT:Seminar zur Analysis SS2002

\documentclass[11pt,leqno,twoside]{article}
\usepackage{a4wide}
\usepackage{german}
\usepackage{amsthm,amsfonts,amssymb,amsmath,upref}
\usepackage[mathscr]{eucal} %fonts $\mathscr{A,B}$ und $\mathcal{A,B}$
\usepackage{amsbsy}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\theoremstyle{plain}
\newtheorem{theorem}{Satz}%[section]
%\newtheorem{theorem*}{Satz}%[section]
\newtheorem{corollary}{Folgerung}%[section]
\newtheorem{lemma}{Lemma}%[section]
\newtheorem{problem}{Problem}%[section]

\theoremstyle{definition}
\newtheorem{definition}{Definition}%[section]
\newtheorem{example}{Beispiel}%[section]
\newtheorem{remark}{Bemerkung}%[section]

%\theoremstyle{remark}
%\newtheorem*{solution}{L\"{o}sung}

\renewcommand{\proofname}{Beweis}
%\renewcommand{\indexname}{Index}

%\numberwithin{equation}%{section}
%\numberwithin{figure}%{section}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\parindent0pt
\parskip0pt
\pagestyle{myheadings}
\renewcommand{\thepage}{\scriptsize\rm{\arabic{page}}}

\newcommand{\ds}{\displaystyle}
\newcommand{\ts}{\textstyle}

\newcommand{\grosso}{\mathcal{O}}
\newcommand{\kleino}{\mbox{\scriptsize$\mathcal{O}$}}
\newcommand{\CDOT}{{\raisebox{-.22ex}{\mbox{\Large$\cdot$}}}}
\newcommand{\DOT}{{\raisebox{-.6ex}{\mbox{\Large$\cdot$}}}}
\newcommand{\AST}{{\raisebox{-.76ex}{\mbox{\Huge$*$}}}}
\newcommand{\wtilde}{{\raisebox{.4ex}{\,$\widetilde{}$\,}}}
\newcommand{\ol}{\overline}
\newcommand{\ul}{\underline}

\newcommand{\cala}{\mathcal{A}}
\newcommand{\calb}{\mathcal{B}}
\newcommand{\cald}{\mathcal{D}}
\newcommand{\calf}{\mathcal{F}}
\newcommand{\calg}{\mathcal{G}}
\newcommand{\calh}{\mathcal{H}}
\newcommand{\calk}{\mathcal{K}}
\newcommand{\call}{\mathcal{L}}
\newcommand{\calm}{\mathcal{M}}
\newcommand{\caln}{\mathcal{N}}
\newcommand{\calr}{\mathcal{R}}
\newcommand{\calt}{\mathcal{T}}

\newcommand{\setC}{\mathbb{C}}
\newcommand{\setN}{\mathbb{N}}
\newcommand{\setP}{\mathbb{P}}
\newcommand{\setQ}{\mathbb{Q}}
\newcommand{\setR}{\mathbb{R}}
\newcommand{\setT}{\mathbb{T}}
\newcommand{\setZ}{\mathbb{Z}}

\newcommand{\om}{\omega}
\newcommand{\ve}{\varepsilon}
\newcommand{\vp}{\varphi}
\newcommand{\vr}{\varrho}
\newcommand{\vt}{\vartheta}
\newcommand{\Alpha}{\mathrm{A}}
\newcommand{\Beta}{\mathrm{B}}
\newcommand{\re}{\mathrm{Re\,}}
\newcommand{\im}{\mathrm{Im\,}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\sgn}{\mathrm{sgn\,}}
\newcommand{\supp}{\mathrm{supp\,}}
\newcommand{\card}{\mathrm{card\,}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\thispagestyle{empty}
\parindent0pt
\parskip0pt

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\textbf{Name, Vorname \hfill SS 2002}                %%%%%%%%%%%%% eintragen!

\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\textbf{Seminar zur Analysis, 1.{} Vortrag:}

\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{center}
\large\textbf{Das Lebesguesche Integrabilit\"{a}tskriterium}
\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{abstract} \noindent
Dieser Vortrag besch\"{a}ftigt sich mit ...
\end{abstract}

\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition} \label{D1}
Eine Menge $N\subseteq\setR$ hei{\ss}t \emph{Nullmenge} oder \emph{Menge vom
Ma{\ss} Null}, wenn es zu jedem $\ve>0$ eine \"{U}berdeckung $\calm$ von $N$
durch h\"{o}chstens abz\"{a}hlbar viele offene Intervalle $I$ mit
\begin{equation} \label{eq1}
\sum_{I\in\calm}\, d(I)<\ve
\end{equation}
gibt. Dabei ist $d(I)$ die L\"{a}nge des Intervalls $I$.
\end{definition}

\begin{corollary}
Jede abz\"{a}hlbare Teilmenge von $\setR$ ist Nullmenge.
\end{corollary}

    \textsf{\small\qquad
    Die Kurzfassung soll allenfalls Beweisideen enthalten.
    Hier nur zur \"{U}bung in \AmS~\LaTeX\,:}

\begin{proof}
Zum Nachweis wird die Nullmengeneigenschaft aus Definition \ref{D1} f\"{u}r
$M=\{m_1,m_2,\ldots\}$ nachgerechnet: Zu $\ve>0$ und $m_k\in M$ sei
$I_k=(m_k-2^{-k-2}\ve,m_k+2^{-k-2}\ve)$ das offene Intervall um $m_k$\,.
Seine L\"{a}nge ist $d(I_k)=2^{-k-1}\ve$\,. Es folgt
\[\sum_{k=1}^\infty\, d(I_k)=\ve\,\sum_{k=1}^\infty\,2^{-k-1}
  =\frac{\ve}{2}<\ve\]
und die Ungleichung \eqref{eq1} ist erf\"{u}llt.
\end{proof}

\begin{theorem}[Integrabilit\"{a}tskriterium von Lebesgue]
Es sei $D=[a,b]$ und $f:D\to\setR$\,. Genau dann ist $f$ auf $D$
Riemann-integrierbar, wenn $f$ auf $D$ beschr\"{a}nkt und fast \"{u}berall
stetig auf $D$ \textup{(}das hei{\ss}t: stetig auf $D\setminus N$ mit einer
Nullmenge $N$\textup{)} ist.
\end{theorem}

Als Anwendung ergeben sich einfache Beweise bekannter S\"{a}tze \"{u}ber
Riemannsche Integrierbarkeit:

\begin{theorem}
Es sei ...
\end{theorem}

\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Beispiel, wie Literatur zitiert wird.
%
%Heuser \cite{Heu2001}, \S\,84, \S\,40 (ohne Aufgaben).
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

    \textsf{\qquad\small
    Ins Literaturverzeichnis geh\"{o}rt nur, was wirklich zitiert wurde,
    etwa in der Form: Artin \cite{Art1964}}

%Mit dem Prozentzeichen kann eine Zeile auskommentiert werden.
%Sie ist dann im dvi file nicht vorhanden.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\renewcommand{\refname}{\normalsize\textbf{Literatur}}
\small
\begin{thebibliography}{9}
\bibitem{Art1964}
Artin, E.:
 \emph{The Gamma Function.}
 Holt, Rinehart and Winston, New York 1964.

\bibitem{HaR1965}
Hardy, G.H. and Rogosinski, W.W.:
 \emph{Fourier Series.}
 3rd ed., Cambridge Univ. Press, Cambridge 1965.

\bibitem{Heu2001}
Heuser, H.:
 \emph{Analysis 1.}
 10.~Auflage. Teubner, Stuttgart 2002 .

\bibitem{Koe1993}
K\"{o}rner, T.W.:
 \emph{Fourier Analysis.}
 Cambridge University Press, Cambridge 1993.

\bibitem{Luc1979}
Lucht, L.G.:
 \emph{Zur Gau{\ss}schen Funktionalgleichung f\"{u}r die Gamma-Funktion auf
 multiplikativen Zahlenmengen.}
 Abh.{} Math.{} Seminar U Hamburg \textbf{49} (1979), 183--188.

\bibitem{Luc2001}
Lucht, L.G.:
 \emph{Analysis I.}
 Vorlesungsausarbeitung, TU Clausthal, WS 2001/02.

\bibitem{Tol1962}
Tolstov, P.T.:
 \emph{Fourier Series.}
 Prentice-Hall Inc., Englewood Cliffs, NJ, 1962.
\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}


























\documentclass[11pt,leqno,twoside]{article}
\usepackage{a4wide}
\usepackage{german}
\usepackage{amsthm,amsfonts,amssymb,amsmath}
\usepackage{amsbsy}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\theoremstyle{plain}
\newtheorem{theorem}{Satz}%[section]
%\newtheorem{theorem*}{Satz}%[section]
\newtheorem{corollary}{Folgerung}%[section]
\newtheorem{lemma}{Lemma}%[section]
\newtheorem{problem}{Problem}%[section]

\theoremstyle{definition}
\newtheorem{definition}{Definition}%[section]
\newtheorem{example}{Beispiel}%[section]
\newtheorem{remark}{Bemerkung}%[section]

%\theoremstyle{remark}
%\newtheorem*{solution}{L\"{o}sung}

\renewcommand{\proofname}{Beweis}
%\renewcommand{\indexname}{Index}

%\numberwithin{equation}%{section}
%\numberwithin{figure}%{section}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\parindent0pt
\parskip0pt
\pagestyle{myheadings}
\renewcommand{\thepage}{\scriptsize\rm{\arabic{page}}}

\newcommand{\ds}{\displaystyle}

\newcommand{\grosso}{\mathcal{O}}
\newcommand{\kleino}{\mbox{\scriptsize$\mathcal{O}$}}
\newcommand{\CDOT}{{\raisebox{-.22ex}{\mbox{\Large$\cdot$}}}}
\newcommand{\DOT}{{\raisebox{-.6ex}{\mbox{\Large$\cdot$}}}}
\newcommand{\AST}{{\raisebox{-.76ex}{\mbox{\Huge$*$}}}}
\newcommand{\wtilde}{{\raisebox{.4ex}{\,$\widetilde{}$\,}}}
\newcommand{\ol}{\overline}
\newcommand{\ul}{\underline}

\newcommand{\cala}{\mathcal{A}}
\newcommand{\calb}{\mathcal{B}}
\newcommand{\cald}{\mathcal{D}}
\newcommand{\calf}{\mathcal{F}}
\newcommand{\calg}{\mathcal{G}}
\newcommand{\calh}{\mathcal{H}}
\newcommand{\calk}{\mathcal{K}}
\newcommand{\call}{\mathcal{L}}
\newcommand{\calm}{\mathcal{M}}
\newcommand{\caln}{\mathcal{N}}
\newcommand{\calr}{\mathcal{R}}
\newcommand{\calt}{\mathcal{T}}

\newcommand{\setC}{\mathbb{C}}
\newcommand{\setN}{\mathbb{N}}
\newcommand{\setP}{\mathbb{P}}
\newcommand{\setQ}{\mathbb{Q}}
\newcommand{\setR}{\mathbb{R}}
\newcommand{\setT}{\mathbb{T}}
\newcommand{\setZ}{\mathbb{Z}}

\newcommand{\om}{\omega}
\newcommand{\ve}{\varepsilon}
\newcommand{\vp}{\varphi}
\newcommand{\vr}{\varrho}
\newcommand{\vt}{\vartheta}
\newcommand{\Alpha}{\mathrm{A}}
\newcommand{\Beta}{\mathrm{B}}
\newcommand{\re}{\mathrm{Re\,}}
\newcommand{\im}{\mathrm{Im\,}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\sgn}{\mathrm{sgn\,}}
\newcommand{\supp}{\mathrm{supp\,}}
\newcommand{\card}{\mathrm{card\,}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\thispagestyle{empty}
\parindent0pt
\parskip0pt

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\textbf{Name, Vorname \hfill SS 2002}                %%%%%%%%%%%%% eintragen!

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
\textbf{Seminar zur Analysis, 1.{} Vortrag:}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
\begin{center}
\large\textbf{Das Lebesguesche Integrabilit\"{a}tskriterium}
\end{center}

\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\textbf{Zusammenfassung:}

\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Der Vortrag behandelt ...