Fubini's theorem FubinisTheorem 2003-05-26 20:33:44 2005-01-08 05:56:16 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{\R}{\mathbb{R}} \PMlinkescapeword{states} \PMlinkescapeword{order} \PMlinkescapeword{time} \textbf{Fubini's theorem} Let $I \subset \R^N$ and $J \subset \R^M$ be compact intervals, and let $f : I \times J \to \R^K$ be a Riemann integrable function such that, for each $x \in I$ the integral \[ F(x) := \int_J f(x, y)\, d\mu_J(y) \] exists. Then $F:I\to\R^K$ is Riemann integrable, and \[ \int_I F = \int_{I\times J} f. \] This theorem effectively states that, given a function of $N$ variables, you may integrate it one variable at a time, and that the order of integration does not affect the result. \textbf{Example} Let $I := [0, \pi/2]\times[0,\pi/2]$, and let $f : I \to \R, x \mapsto \sin(x)\cos(y)$ be a function. Then \begin{equation*} \begin{split} \int_I f &= \iint_{[0, \pi/2]\times[0,\pi/2]} \sin(x)\cos(y) \\ &=\int_0^{\pi/2} \left( \int_0^{\pi/2} \sin(x)\cos(y)\,dy\right)\,dx \\ &=\int_0^{\pi/2} \sin(x)\left(1 - 0\right)\,dx =(0 - -1) = 1. \end{split} \end{equation*} Note that it is often simpler (and no less correct) to write $\idotsint_I f$ as $\int_I f$.