Fatou-Lebesgue 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 $|f_{n}|\leq\Phi$ for each $n$ , then

$g=\liminf_{n\rightarrow\infty}f_{n}\;\;\textnormal{and}\;h=\limsup_{n% \rightarrow\infty}f_{n}$

are both integrable, and

$-\infty<\int gd\mu\leq\liminf_{n\rightarrow\infty}\int f_{n}d\mu\leq\limsup_{k% \rightarrow\infty}\int f_{n}d\mu\leq\int hd\mu<\infty.$

Title	Fatou-Lebesgue theorem
Canonical name	FatouLebesgueTheorem
Date of creation	2013-03-22 13:12:53
Last modified on	2013-03-22 13:12:53
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	7
Author	Koro (127)
Entry type	Theorem
Classification	msc 28A20
Related topic	FatousLemma