Product Measures and Fubini - Core Definitions

Product measures provide the mathematical framework for defining integration over products of spaces, such as $\mathbb{R}^n = \mathbb{R} \times \cdots \times \mathbb{R}$ . This construction is essential for multivariable integration and probability theory.

DefinitionProduct Sigma-Algebra

Let $(X, \mathcal{F})$ and $(Y, \mathcal{G})$ be measurable spaces. The product sigma-algebra $\mathcal{F} \otimes \mathcal{G}$ on $X \times Y$ is the sigma-algebra generated by the measurable rectangles: $\mathcal{F} \otimes \mathcal{G} = \sigma(\{A \times B : A \in \mathcal{F}, B \in \mathcal{G}\})$

A set of the form $A \times B$ is called a measurable rectangle.

The product sigma-algebra is the smallest sigma-algebra making all rectangles measurable. It contains countable unions and intersections of rectangles, but not all subsets of $X \times Y$ .

DefinitionProduct Measure

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

The existence and uniqueness follow from Caratheodory's Extension Theorem, applied to the algebra of finite unions of rectangles.

ExampleLebesgue Measure on $\mathbb{R}^2$

Let $\lambda$ denote Lebesgue measure on $\mathbb{R}$ . Then $\lambda \otimes \lambda$ is Lebesgue measure on $\mathbb{R}^2$ .

For a rectangle $[a, b] \times [c, d]$ : $(\lambda \otimes \lambda)([a, b] \times [c, d]) = (b - a)(d - c)$

This is the usual formula for the area of a rectangle, showing that product measure generalizes geometric area.

Remark

Important observations:

$\sigma$ -finiteness is essential: Without $\sigma$ -finiteness, the product measure may not be unique. There can be multiple measures agreeing on rectangles but differing on other sets.
Sections: For $E \subseteq X \times Y$ and $x \in X$ , the $x$ -section is $E_x = \{y \in Y : (x, y) \in E\}$ . Similarly, $E^y = \{x \in X : (x, y) \in E\}$ is the $y$ -section.

If $E \in \mathcal{F} \otimes \mathcal{G}$ , then $E_x \in \mathcal{G}$ for all $x$ , and $E^y \in \mathcal{F}$ for all $y$ . The converse is false.

Generalization: Product measures extend to finite products: $\mu_1 \otimes \cdots \otimes \mu_n$ on $X_1 \times \cdots \times \times_n$ .