Fubini and Tonelli Theorems

8.1 Product Measures

Let $(X, \mathcal{F}, \mu)$ and $(Y, \mathcal{G}, \nu)$ be $\sigma$-finite measure spaces. The product $\sigma$-algebra is $\mathcal{F} \otimes \mathcal{G} = \sigma(\{A \times B : A \in \mathcal{F},\ B \in \mathcal{G}\})$.

Theorem 8.1 (Existence of Product Measure). There exists a unique measure $\mu \times \nu$ on $\mathcal{F} \otimes \mathcal{G}$ such that

$(\mu \times \nu)(A \times B) = \mu(A) \cdot \nu(B)$

for all $A \in \mathcal{F}$ and $B \in \mathcal{G}$.

8.2 Tonelli”s Theorem

Theorem 8.2 (Tonelli). If $f : X \times Y \to [0, \infty]$ is $\mathcal{F} \otimes \mathcal{G}$-measurable, then:

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

8.3 Fubini’s Theorem

Theorem 8.3 (Fubini). If $f \in L^1(\mu \times \nu)$, then for a.e. $x \in X$, $f(x, \cdot) \in L^1(\nu)$; for a.e. $y \in Y$, $f(\cdot, y) \in L^1(\mu)$; and

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

Caution. The order of integration matters when $f$ is not integrable. For example, the function $f(x, y) = (x^2 - y^2)/(x^2 + y^2)^2$ on $[0,1]^2$ has different iterated integrals.

8.4 Worked Example

Problem. Compute $\int_0^\infty \int_0^\infty e^{-(x^2 + y^2)}\, dy\, dx$ using Fubini-Tonelli.

Solution. By Tonelli’s theorem (since $e^{-(x^2+y^2)} \geq 0$):

$\int_0^\infty \int_0^\infty e^{-(x^2 + y^2)}\, dy\, dx = \int_0^\infty e^{-x^2}\, dx \cdot \int_0^\infty e^{-y^2}\, dy = \left(\frac{\sqrt{\pi}}{2}\right)^2 = \frac{\pi}{4}$

$\blacksquare$