# proof of Fubini’s theorem for the Lebesgue integral

Let ${\mu}_{x}$ and ${\mu}_{y}$ be measures^{} on $X$ and $Y$ respectively, let $\mu $ be the product measure^{} ${\mu}_{x}\otimes {\mu}_{y}$, and let $f(x,y)$ be $\mu $-integrable on $A\subset X\times Y$. Then

$${\int}_{A}f(x,y)\mathit{d}\mu ={\int}_{X}\left({\int}_{{A}_{x}}f(x,y)\mathit{d}{\mu}_{y}\right)\mathit{d}{\mu}_{x}={\int}_{Y}\left({\int}_{{A}_{y}}f(x,y)\mathit{d}{\mu}_{x}\right)\mathit{d}{\mu}_{y}$$ |

where

$${A}_{x}=\{y\mid (x,y)\in A\},{A}_{y}=\{x\mid (x,y)\in A\}$$ |

Proof: Assume for now that $f(x,y)\ge 0$. Consider the set

$$U=X\times Y\times \mathbb{R}$$ |

equipped with the measure

$${\mu}_{u}={\mu}_{x}\otimes {\mu}_{y}\otimes {\mu}^{1}=\mu \otimes {\mu}^{1}={\mu}_{x}\otimes \lambda $$ |

where ${\mu}^{1}$ is ordinary Lebesgue measure^{} and $\lambda ={\mu}_{y}\otimes {\mu}^{1}$. Also consider the set $W\subset U$ defined by

$$W=\{(x,y,z)\mid (x,y)\in A,0\le z\le f(x,y)\}$$ |

Then

$${\mu}_{u}\left(W\right)={\int}_{A}f(x,y)\mathit{d}\mu $$ |

And

$${\mu}_{u}\left(W\right)={\int}_{X}\lambda \left({W}_{x}\right)\mathit{d}{\mu}_{x}$$ |

where

$${W}_{x}=\{(y,z)\mid (x,y,z)\in W\}$$ |

However, we also have that

$$\lambda \left({W}_{x}\right)={\int}_{{A}_{x}}f(x,y)\mathit{d}{\mu}_{y}$$ |

Combining the last three equations gives us Fubini’s theorem. To remove the restriction^{} that $f(x,y)$ be nonnegative, write $f$ as

$$f(x,y)={f}^{+}(x,y)-{f}^{-}(x,y)$$ |

where

$${f}^{+}(x,y)=\frac{|f(x,y)|+f(x,y)}{2},{f}^{-}(x,y)=\frac{|f(x,y)|-f(x,y)}{2}$$ |

are both nonnegative.

Title | proof of Fubini’s theorem for the Lebesgue integral^{} |
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Canonical name | ProofOfFubinisTheoremForTheLebesgueIntegral |

Date of creation | 2013-03-22 15:21:52 |

Last modified on | 2013-03-22 15:21:52 |

Owner | azdbacks4234 (14155) |

Last modified by | azdbacks4234 (14155) |

Numerical id | 4 |

Author | azdbacks4234 (14155) |

Entry type | Proof |

Classification | msc 28A35 |