Fubini’s theorem

Fubini’s theorem Let $I\subset\mathbb{R}^{N}$ and $J\subset\mathbb{R}^{M}$ be compact intervals, and let $f:I\times J\to\mathbb{R}^{K}$ 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 $F:I\to\mathbb{R}^{K}$ 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.

Example Let $I:=[0,\pi/2]\times[0,\pi/2]$, and let $f:I\to\mathbb{R},x\mapsto\sin(x)\cos(y)$ be a function. Then

 $\begin{split}\displaystyle\int_{I}f&\displaystyle=\iint_{[0,\pi/2]\times[0,\pi% /2]}\sin(x)\cos(y)\\ &\displaystyle=\int_{0}^{\pi/2}\left(\int_{0}^{\pi/2}\sin(x)\cos(y)\,dy\right)% \,dx\\ &\displaystyle=\int_{0}^{\pi/2}\sin(x)\left(1-0\right)\,dx=(0--1)=1.\end{split}$

Note that it is often simpler (and no less correct) to write $\idotsint_{I}f$ as $\int_{I}f$.

Title Fubini’s theorem FubinisTheorem 2013-03-22 13:39:13 2013-03-22 13:39:13 mathcam (2727) mathcam (2727) 11 mathcam (2727) Theorem msc 26B12 TonellisTheorem FubinisTheoremForTheLebesgueIntegral IntegrationUnderIntegralSign