# Tonelli's Theorem

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## Theorem

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

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

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

Then:

- $\ds \int_{X \times Y} f \map \rd {\mu \times \nu} = \int_Y \int_X \map f {x, y} \map {\d \mu} x \map {\d \nu} y = \int_X \int_Y \map f {x, y} \map {\d \nu} y \map {\d \mu} x$

## Proof

## Also see

- Fubini's Theorem, a similar result pertaining to $\mu$-integrable functions.

## Source of Name

This entry was named for Leonida Tonelli.

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $13.8$