Product measure

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 . The product sigma-algebra $\mathcal{A}\otimes\mathcal{B}$ is the sigma-algebra on $X\times Y$ generated by measurable rectangles $A\times B$ with $A\in\mathcal{A}$ and $B\in\mathcal{B}$. A measure $\mu\otimes\nu$ on $(X\times Y,\mathcal{A}\otimes\mathcal{B})$ is called a product measure if

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 $\mathbb{R}$, then $\lambda\otimes\lambda$ is the standard Lebesgue measure on $\mathbb{R}^2$ (and more generally $\lambda^{\otimes n}$ gives Lebesgue measure on $\mathbb{R}^n$).
• 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 $(\mu\otimes\nu)(E)=|E|$.