|
|
|
Viewing Version
4
of
'Vitali convergence theorem'
|
[ view 'Vitali convergence theorem'
|
back to history
]
| Title of object: |
Vitali convergence theorem |
| Canonical Name: |
VitaliConvergenceTheorem |
| Type: |
Theorem |
| Created on: |
2006-09-27 21:27:35 |
| Modified on: |
2006-10-06 09:03:27 |
| Classification: |
msc:28A20 |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{enumerate}
%\usepackage{graphicx}
%\usepackage{psfrag}
%\usepackage{xypic}
% define commands here
\newcommand{\complex}{\mathbb{C}}
\newcommand{\real}{\mathbb{R}}
\newcommand{\rat}{\mathbb{Q}}
\newcommand{\nat}{\mathbb{N}}
\newcommand{\Le}{\mathbf{L}}
\providecommand{\abs}[1]{\lvert#1\rvert}
\providecommand{\absW}[1]{\left\lvert#1\right\rvert}
\providecommand{\absB}[1]{\Bigl\lvert#1\Bigr\rvert}
\providecommand{\norm}[1]{\lVert#1\rVert}
\providecommand{\normW}[1]{\left\lVert#1\right\rVert}
\providecommand{\normB}[1]{\Bigl\lVert#1\Bigr\rVert}
|
Content:
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{Remark}
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.
In a finite measure space, condition (iii) is trivial.
Indeed, 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;
nevertheless, 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}
|
|
|
|
|
|
