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# 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

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

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

Related:

TonellisTheorem, FubinisTheoremForTheLebesgueIntegral, IntegrationUnderIntegralSign

Type of Math Object:

Theorem

Major Section:

Reference

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## Mathematics Subject Classification

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