modes of convergence of sequences of measurable functions
Let $(X,\U0001d505,\mu )$ be a measure space^{}, ${f}_{n}:X\to [\mathrm{\infty},\mathrm{\infty}]$ be measurable functions^{} for every positive integer $n$, and $f:X\to [\mathrm{\infty},\mathrm{\infty}]$ be a measurable function. The following are modes of convergence of $\{{f}_{n}\}$:

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$\{{f}_{n}\}$ converges almost everywhere to $f$ if $\mu \left(X\{x\in X:\underset{n\to \mathrm{\infty}}{lim}{f}_{n}(x)=f(x)\}\right)=0$

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$\{{f}_{n}\}$ converges almost uniformly to $f$ if, for every $\epsilon >0$, there exists ${E}_{\epsilon}\in \U0001d505$ with $$ and $\{{f}_{n}\}$ converges uniformly to $f$ on ${E}_{\epsilon}$

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$\{{f}_{n}\}$ converges in measure to $f$ if, for every $\epsilon >0$, there exists a positive integer $N$ such that, for every positive integer $n\ge N$, $$.

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If, in , $f$ and each ${f}_{n}$ are also Lebesgue integrable^{}, $\{{f}_{n}\}$ converges in ${L}^{\mathrm{1}}\mathit{}\mathrm{(}\mu \mathrm{)}$ to $f$ if $\underset{n\to \mathrm{\infty}}{lim}{\displaystyle {\int}_{X}}\left{f}_{n}f\right\mathit{d}\mu =0$.
A lot of theorems in real analysis (http://planetmath.org/BibliographyForRealAnalysis) deal with these modes of convergence. For example, Fatou’s lemma, Lebesgue’s monotone convergence theorem^{}, and Lebesgue’s dominated convergence theorem give conditions on sequences^{} of measurable functions that converge almost everywhere under which they also converge in ${L}^{1}(\mu )$. Also, Egorov’s theorem that, if $$, then convergence almost everywhere implies almost uniform convergence^{}.
Title  modes of convergence of sequences of measurable functions 

Canonical name  ModesOfConvergenceOfSequencesOfMeasurableFunctions 
Date of creation  20130322 16:14:05 
Last modified on  20130322 16:14:05 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  7 
Author  Wkbj79 (1863) 
Entry type  Definition 
Classification  msc 28A20 
Related topic  TravelingHumpSequence 
Related topic  VitaliConvergenceTheorem 
Defines  converges almost everywhere 
Defines  convergence almost everywhere 
Defines  converges almost uniformly 
Defines  almost uniform convergence 
Defines  converges in measure 
Defines  convergence in measure 
Defines  converges in ${L}^{1}(\mu )$ 
Defines  ${L}^{1}(\mu )$ convergence 