Tonelli's Theorem

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

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

Let $f: X \times Y \to \overline{\R}_{\ge 0}$ be a positive $\Sigma_1 \otimes \Sigma_2$-measurable function.

Then:


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

where $\mu \times \nu$ is the product measure of $\mu$ and $\nu$.

Also see

 * Fubini's Theorem, a very similar result pertaining to integrable functions.