Product measure - Knowlpedia

A product measure combines two measures into a measure on a Cartesian product. Let $(X,\mathcal{A},\mu)$ and $(Y,\mathcal{B},\nu)$ be measure spaces. Their product sigma-algebra is denoted by $\mathcal A\otimes\mathcal B$ . A measure $\mu\otimes\nu$ on $(X\times Y,\mathcal{A}\otimes\mathcal{B})$ is called a product measure if

(\mu\otimes\nu)(A\times B)=\mu(A)\,\nu(B) \quad\text{for all }A\in\mathcal{A},\ B\in\mathcal{B}.

When $\mu$ and $\nu$ are $\sigma$ -finite, such a measure exists and is uniquely determined by its values on rectangles; it is the measure used in Tonelli's theorem and Fubini's theorem for iterated integration.

Examples

If $\lambda$ is Lebesgue measure on $\mathbb{R}$ , the completion of $\lambda\otimes\lambda$ is Lebesgue measure on $\mathbb{R}^2$ ; the product sigma-algebra itself need not be complete. The analogous completion gives Lebesgue measure in higher dimensions.
If $\mu$ and $\nu$ are counting measures on $\mathbb{N}$ , then $\mu\otimes\nu$ is counting measure on $\mathbb{N}\times\mathbb{N}$ : for any finite set $E\subset \mathbb{N}\times\mathbb{N}$ one has $(\mu\otimes\nu)(E)=|E|$ .