Fatou-Lebesgue theorem FatouLebesgueTheorem 2002-12-07 10:48:41 2004-11-27 13:55:06 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 \[-\infty < \int g d\mu\leq \liminf_{n\rightarrow\infty}\int f_nd\mu\leq \limsup_{k\rightarrow\infty}\int f_n d\mu\leq \int h d\mu < \infty.\]