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Fatou-Lebesgue theorem (Theorem)

Let $ (X,\mu)$ be a measure space. If $ \Phi\colon X\to \mathbb{R}$ is a nonnegative function with $ \int \Phi d\mu <\infty$, and if $ f_1, f_2,\dots$ is a sequence of measurable functions such that $ \vert f_n\vert\leq \Phi$ for each $ n$, then

$\displaystyle g=\liminf_{n\rightarrow\infty} f_n \;\;\textnormal{and}\; h=\limsup_{n\rightarrow\infty} f_n$
are both integrable, and
$\displaystyle -\infty < \int g d\mu\leq \liminf_{n\rightarrow\infty}\int f_nd\mu\leq \limsup_{k\rightarrow\infty}\int f_n d\mu\leq \int h d\mu < \infty.$



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


Attachments:
proof of Fatou-Lebesgue theorem (Proof) by Wkbj79
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Cross-references: measurable functions, sequence, function, measure space
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This is version 4 of Fatou-Lebesgue theorem, born on 2002-12-07, modified 2004-11-27.
Object id is 3679, canonical name is FatouLebesgueTheorem.
Accessed 6163 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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