PlanetMath: modes of convergence of sequences of measurable functions

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modes of convergence of sequences of measurable functions

(Definition)

Let $(X,\mathfrak{B},\mu)$ be a measure space, $f_n \colon X \to [-\infty, \infty]$ be measurable functions for every positive integer , and $f \colon X \to [-\infty, \infty]$ be a measurable function. The following are modes of convergence of $\{f_n\}$ :

$\{f_n\}$ converges almost everywhere to if $\displaystyle \mu \left( X-\{x \in X: \lim_{n \to \infty} f_n(x)=f(x)\} \right)=0$
$\{f_n\}$ converges almost uniformly to if, for every $\varepsilon >0$ , there exists $E_\varepsilon \in \mathfrak{B}$ with $\mu (X-E_\varepsilon) <\varepsilon$ and $\{f_n\}$ converges uniformly to on $E_\varepsilon$
$\{f_n\}$ converges in measure to if, for every $\varepsilon >0$ , there exists a positive integer such that, for every positive integer $n \ge N$ , $\displaystyle \mu \left( \{ x \in X:\vert f_n(x)-f(x)\vert\ge \varepsilon \} \right)<\varepsilon$ .
If, in addition, and each are also Lebesgue integrable, $\{f_n\}$ converges in $L^1(\mu)$ to if $\displaystyle \lim_{n \to \infty} \int_X \left\vert f_n-f \right\vert \, d\mu =0$ .

A lot of theorems in real analysis 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 states that, if $\mu(X)<\infty$ , then convergence almost everywhere implies almost uniform convergence.

"modes of convergence of sequences of measurable functions" is owned by Wkbj79.

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Also defines:

converges almost everywhere, convergence almost everywhere, converges almost uniformly, almost uniform convergence, converges in measure, convergence in measure, converges in $L^1(\mu)$ , $L^1(\mu)$ convergence

Attachments:: Vitali convergence theorem (Theorem) by stevecheng

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Cross-references: implies, Egorov's theorem, almost everywhere, sequences, Lebesgue's dominated convergence theorem, Lebesgue's monotone convergence theorem, Fatou's lemma, Lebesgue integrable, converges uniformly, integer, positive, measurable functions, measure space
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This is version 4 of modes of convergence of sequences of measurable functions, born on 2006-09-10, modified 2007-04-15.
Object id is 8335, canonical name is ModesOfConvergenceOfSequencesOfMeasurableFunctions.
Accessed 3135 times total.

Classification:

AMS MSC:

28A20 (Measure and integration :: Classical measure theory :: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence)

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