Definition:Product Measure

Definition
Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be $\sigma$-finite measure spaces.

Let $\struct {X \times Y, \Sigma_X \otimes \Sigma_Y}$ be the product measurable space of $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$.

The product measure of $\mu$ and $\nu$, denoted $\mu \times \nu$, is the unique measure with:


 * $\forall E_1 \in \Sigma_X, E_2 \in \Sigma_Y: \map {\paren {\mu \times \nu} } {E_1 \times E_2} = \map \mu {E_1} \map \nu {E_2}$

That this is defines a unique measure on $\Sigma_X \otimes \Sigma_Y$ is shown in Uniqueness of Product Measures.

From Existence of Product Measures, the measure is explicitly given by:


 * $\ds \map {\paren {\mu \times \nu} } E = \int_X \map \nu {E_x} \rd \mu = \int_Y \map \mu {E^y} \rd \nu$

for each $E \in \Sigma_X \otimes \Sigma_Y$, where:


 * $E^x$ is the $x$-vertical section of $E$
 * $E^y$ is the $y$-horizontal section of $E$.