Vitali convergence theoremVitaliConvergenceTheorem2006-09-27 21:27:352006-10-06 09:45:50Theoremmodes of convergence of sequences of measurable functions\usepackage{amssymb}
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Let $f_1, f_2, \dotsc$ be $\Le^p$-integrable functions on some measure space, for $1 \leq p < \infty$.
The sequence $\{ f_n \}$ converges in $\Le^p$ to a measurable function $f$
if and and only if
\begin{enumerate}[i]
\item
the sequence $\{ f_n \}$ converges to $f$ in measure;
\item
the functions $\{ \abs{f_n}^p \}$ are uniformly integrable; and
\item
for every $\epsilon > 0$, there exists a set $E$
of finite measure, such that $\int_{E^\mathrm{c}} \abs{f_n}^p < \epsilon$
for all $n$.
\end{enumerate}
\subsection*{Remarks}
This theorem can be used as a replacement for the more
well-known dominated convergence theorem, when a
dominating \PMlinkescapetext{factor} cannot be found for the functions
$f_n$ to be integrated.
(If this theorem is known, the dominated convergence theorem
can be derived as a special case.)
In a finite measure space, condition (iii) is trivial.
In fact, condition (iii) is the tool used to reduce considerations
in the general case to the case of a finite measure space.
In probability \PMlinkescapetext{theory}, the definition of ``uniform integrability''
is slightly different from its definition in general measure theory;
either definition may be used in the statement of this theorem.
\begin{thebibliography}{3}
\bibitem{Folland}
Gerald B. Folland. {\it Real Analysis: Modern Techniques and Their Applications}, second ed. Wiley-Interscience, 1999.
\bibitem{Rosenthal}
Jeffrey S. Rosenthal. {\it A First Look at Rigorous Probability Theory}.
World Scientific, 2003.
\end{thebibliography}