# 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

$$F(x):={\int}_{J}f(x,y)\mathit{d}{\mu}_{J}(y)$$ |

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 $f:I\to \mathbb{R},x\mapsto \mathrm{sin}(x)\mathrm{cos}(y)$ be a function. Then

$$\begin{array}{cc}\hfill {\int}_{I}f& ={\iint}_{[0,\pi /2]\times [0,\pi /2]}\mathrm{sin}(x)\mathrm{cos}(y)\hfill \\ & ={\int}_{0}^{\pi /2}\left({\int}_{0}^{\pi /2}\mathrm{sin}(x)\mathrm{cos}(y)\mathit{d}y\right)\mathit{d}x\hfill \\ & ={\int}_{0}^{\pi /2}\mathrm{sin}(x)(1-0)dx=(0--1)=1.\hfill \end{array}$$ |

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

Title | Fubini’s theorem |
---|---|

Canonical name | FubinisTheorem |

Date of creation | 2013-03-22 13:39:13 |

Last modified on | 2013-03-22 13:39:13 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 11 |

Author | mathcam (2727) |

Entry type | Theorem |

Classification | msc 26B12 |

Related topic | TonellisTheorem |

Related topic | FubinisTheoremForTheLebesgueIntegral |

Related topic | IntegrationUnderIntegralSign |