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 $E$.

Product Measure Space

The product of $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$ is the measure space:

$\left({X \times Y, \Sigma_1 \otimes \Sigma_2, \mu \times \nu}\right)$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $13.6$