Fatou's lemma FatousLemma 2002-12-07 10:35:16 2002-12-07 11:33:20 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 If $f_1, f_2,\dots$ is a sequence of nonnegative measurable functions in a measure space $X$, then \[ \int_X \liminf_{n\rightarrow\infty} f_n \leq \liminf_{n\rightarrow\infty}\int_X f_n \]