Product Measures and Fubini - Key Properties

The fundamental property of product measures is that integration can be computed via iterated integrals. This is the content of Fubini's and Tonelli's theorems, which justify interchanging the order of integration.

TheoremTonelli's Theorem

Let $(X, \mathcal{F}, \mu)$ and $(Y, \mathcal{G}, \nu)$ be $\sigma$ -finite measure spaces. If $f: X \times Y \to [0, \infty]$ is measurable with respect to $\mathcal{F} \otimes \mathcal{G}$ , then:

For almost every $x \in X$ , the function $y \mapsto f(x, y)$ is $\mathcal{G}$ -measurable
The function $x \mapsto \int_Y f(x, y) \, d\nu(y)$ is $\mathcal{F}$ -measurable
$\int_{X \times Y} f \, d(\mu \otimes \nu) = \int_X \left(\int_Y f(x, y) \, d\nu(y)\right) d\mu(x) = \int_Y \left(\int_X f(x, y) \, d\mu(x)\right) d\nu(y)$

Tonelli's theorem applies to non-negative functions, requiring no integrability conditions beyond measurability.

ExampleComputing Double Integrals

To compute $\int_{[0,1] \times [0,1]} e^{x+y} \, d(\lambda \otimes \lambda)$ , we use Tonelli: $\int_0^1 \int_0^1 e^{x+y} \, dy \, dx = \int_0^1 e^x \left(\int_0^1 e^y \, dy\right) dx = \int_0^1 e^x (e - 1) \, dx = (e - 1)^2$

Alternatively, integrating in the opposite order gives the same result: $\int_0^1 \int_0^1 e^{x+y} \, dx \, dy = (e - 1)^2$

Remark

Key aspects of Tonelli's Theorem:

Non-negativity is crucial: For non-negative functions, the theorem always applies. The integral might be infinite, but the iterated integrals exist and agree.
Measurability of sections: The theorem guarantees that for a.e. $x$ , the section $f(x, \cdot)$ is measurable, even though this is not obvious from the measurability of $f$ on the product.
Order of integration: Both orders of integration yield the same value. This symmetry is a consequence of the uniqueness of product measure.
Application to non-negative measurable functions: Tonelli is often used to verify integrability: if either iterated integral is finite, then $f$ is integrable with respect to the product measure.
Extension to complete products: The theorem extends to completions of product measures, which is important for Lebesgue measure on $\mathbb{R}^n$ .

Tonelli's Theorem is particularly useful for checking integrability and computing integrals of non-negative functions, as it requires no a priori integrability assumptions.