\begin{definition}[Definition:Product Measure]
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 $E$.
\end{definition}