Fubini's Theorem

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Theorem

Let $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$ be $\sigma$-finite measure spaces.

Let $\left({X \times Y, \Sigma_1 \otimes \Sigma_2, \mu \times \nu}\right)$ be the product measure space of $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$.

Let $f: X \times Y \to \R$ be a $\Sigma_1 \otimes \Sigma_2$-measurable function.


Suppose that:

$\displaystyle \int_{X \times Y} \left\vert{f}\right\vert \, \mathrm d \left({\mu \times \nu}\right) < \infty$


Then $f$ is $\mu \times \nu$-integrable, and:

$\displaystyle \int_{X \times Y} f \, \mathrm d \left({\mu \times \nu}\right) = \int_Y \int_X f \left({x, y}\right) \, \mathrm d \mu \left({x}\right) \, \mathrm d \nu \left({y}\right) = \int_X \int_Y f \left({x, y}\right) \, \mathrm d \nu \left({y}\right) \, \mathrm d \mu \left({x}\right)$


Proof


Also see


Source of Name

This entry was named for Guido Fubini.


Sources