modes of convergence of sequences of measurable functions

Let $(X,\mathfrak{B},\mu)$ be a measure space, $f_{n}\colon X\to[-\infty,\infty]$ be measurable functions for every positive integer $n$ , and $f\colon X\to[-\infty,\infty]$ 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 $\displaystyle\mu\left(X-\{x\in X:\lim_{n\to\infty}f_{n}(x)=f(x)\}\right)=0$
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$\{f_{n}\}$ converges almost uniformly to $f$ if, for every $\varepsilon>0$ , there exists $E_{\varepsilon}\in\mathfrak{B}$ with $\mu(X-E_{\varepsilon})<\varepsilon$ and $\{f_{n}\}$ converges uniformly to $f$ on $E_{\varepsilon}$
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$\{f_{n}\}$ converges in measure to $f$ if, for every $\varepsilon>0$ , there exists a positive integer $N$ such that, for every positive integer $n\geq N$ , $\displaystyle\mu\left(\{x\in X:|f_{n}(x)-f(x)|\geq\varepsilon\}\right)<\varepsilon$ .
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If, in , $f$ and each $f_{n}$ are also Lebesgue integrable, $\{f_{n}\}$ converges in $L^{1}(\mu)$ to $f$ if $\displaystyle\lim_{n\to\infty}\int_{X}\left|f_{n}-f\right|\,d\mu=0$ .

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 $\mu(X)<\infty$ , then convergence almost everywhere implies almost uniform convergence.

Title	modes of convergence of sequences of measurable functions
Canonical name	ModesOfConvergenceOfSequencesOfMeasurableFunctions
Date of creation	2013-03-22 16:14:05
Last modified on	2013-03-22 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