%AUTOR:Lucht CONT:Seminar zur Analysis SS2002

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

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

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

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

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\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}

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

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\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}

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

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

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

\medskip

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\textbf{Zusammenfassung:}

\medskip

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Der Vortrag behandelt ...


