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Fatou's lemma (Theorem)

If $ f_1, f_2,\dots$ is a sequence of nonnegative measurable functions in a measure space $ X$, then

$\displaystyle \int_X \liminf_{n\rightarrow\infty} f_n \leq \liminf_{n\rightarrow\infty}\int_X f_n $



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


Attachments:
proof of Fatou's lemma (Proof) by paolini
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Cross-references: measure space, measurable functions, sequence
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This is version 3 of Fatou's lemma, born on 2002-12-07, modified 2002-12-07.
Object id is 3678, canonical name is FatousLemma.
Accessed 15530 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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