Fubini's Theorem

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:


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

Also see

 * Tonelli's Theorem