dominated convergence theorem DominatedConvergenceTheorem 2002-12-07 09:55:22 2009-06-11 13:21:12 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 $X$ be a measure space, and let $\Phi,f_1,f_2,\dots$ be measurable functions such that $\int_X \Phi <\infty$ and $|f_n|\leq \Phi$ for each $n$. If $f_n\rightarrow f$ almost everywhere, then $f$ is integrable and \[ \lim_{n\rightarrow\infty} \int_X f_n = \int_X f. \] This theorem is a corollary of the Fatou-Lebesgue theorem.