User:Caliburn/s/mt/Uniqueness of Product Measures/Corollary

Corollary to Uniqueness of Product Measures

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

Let $\GG_X$ and $\GG_Y$ be generators for $\Sigma_X$ and $\Sigma_Y$, respectively.

Then the unique measure on the product space $\struct {X \times Y, \Sigma_X \otimes \Sigma_Y}$ such that:

$\forall G_1 \in \GG_X, G_2 \in \GG_Y: \map \rho {G_1 \times G_2} = \map \mu {G_1} \, \map \nu {G_2}$

is given by:

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

for each $E \in \Sigma_X \otimes \Sigma_Y$.

Proof

From Existence of Product Measures, the measures $\rho_1$, $\rho_2$ given by:

$\ds \map {\rho_1} E = \int_X \map \nu {E_x} \rd \mu$

and:

$\ds \map {\rho_2} E = \int_Y \map \mu {E^y} \rd \nu$

for each $E \in \Sigma_X \otimes \Sigma_Y$ are such that:

$\forall G_1 \in \GG_X, G_2 \in \GG_Y: \map \rho {G_1 \times G_2} = \map \mu {G_1} \, \map \nu {G_2}$

From Uniqueness of Product Measures, we have:

$\rho_1 = \rho_2 = \mu \times \nu$

Hence the result.