monotone convergence theorem MonotoneConvergenceTheorem 2002-06-14 04:06:26 2007-04-22 13:33:42 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 \newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}} \newcommand{\Expect}{\mathbb{E}} \newcommand{\norm}[1]{\left\|#1\right\|} % Some sets \newcommand{\Nats}{\mathbb{N}} \newcommand{\Ints}{\mathbb{Z}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} %%%%%% END OF SAVED PREAMBLE %%%%%% Let $X$ be a measure space, and let $0\leq f_1\leq f_2\leq\cdots$ be a monotone increasing sequence of nonnegative measurable functions. Let $f\colon X \to \mathbb{R}\cup \{\infty\}$ be the function defined by $f(x) = \lim_{n\rightarrow\infty} f_n(x)$. Then $f$ is measurable, and $$\lim_{n\rightarrow\infty} \int_X f_n = \int_X f.$$ \textbf{Remark.} This theorem is the first of several theorems which allow us to ``exchange integration and limits''. It requires the use of the Lebesgue integral: with the Riemann integral, we cannot even formulate the theorem, lacking, as we do, the concept of ``almost everywhere''. For instance, the characteristic function of the rational numbers in $[0,1]$ is not Riemann integrable, despite being the limit of an increasing sequence of Riemann integrable functions.