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dominated convergence theorem (Theorem)

Let $ X$ be a measure space, and let $ \Phi,f_1,f_2,\dots$ be measurable functions such that $ \int_X \Phi <\infty$ and $ \vert f_n\vert\leq \Phi$ for each $ n$. If $ f_n\rightarrow f$ almost everywhere, then $ f$ is integrable and

$\displaystyle \lim_{n\rightarrow\infty} \int_X f_n = \int_X f. $

This theorem is a corollary of the Fatou-Lebesgue theorem.

A possible generalization is that if $ \{f_r: r\in \mathbb{R}\}$ is a family of measurable functions such that $ \vert f_r\vert\leq \vert\Phi\vert$ for each $ r\in \mathbb{R}$ and $ f_r\xrightarrow[r\rightarrow 0]{} f$, then $ f$ is integrable and

$\displaystyle \lim_{r\rightarrow 0} \int_X f_r = \int_X f. $



"dominated convergence theorem" is owned by Koro.
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See Also: monotone convergence theorem, Fatou's lemma, Vitali convergence theorem

Other names:  Lebesgue's dominated convergence theorem

Attachments:
proof of dominated convergence theorem (Proof) by paolini
proof of dominated convergence theorem (Proof) by rspuzio
criterion for interchanging summation and integration (Result) by rspuzio
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Cross-references: Fatou-Lebesgue theorem, almost everywhere, measurable functions, measure space
There are 11 references to this entry.

This is version 9 of dominated convergence theorem, born on 2002-12-07, modified 2004-10-15.
Object id is 3677, canonical name is DominatedConvergenceTheorem.
Accessed 12428 times total.

Classification:
AMS MSC: 28A20 (Measure and integration :: Classical measure theory :: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence)
Pending Errata and Addenda
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