# 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 FatouLebesgueTheorem 2013-03-22 13:12:53 2013-03-22 13:12:53 Koro (127) Koro (127) 7 Koro (127) Theorem msc 28A20 FatousLemma