Uniqueness of Product Measures

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Let $\struct {X, \Sigma_1, \mu}$ and $\struct {Y, \Sigma_2, \nu}$ be measure spaces.

Let $\GG_1$ and $\GG_2$ be generators for $\Sigma_1$ and $\Sigma_2$, respectively.

Suppose that $\GG_1$ and $\GG_2$ are closed under intersection.

Suppose further that there are exhausting sequences $\sequence {G_{1, n} }_{n \mathop \in \N}$ and $\sequence {G_{2, n} }_{n \mathop \in \N}$ in $\GG_1$ and $\GG_2$, respectively, such that:

$\forall n \in \N: \map \mu {G_{1, n} } < \infty$
$\forall n \in \N: \map \nu {G_{2, n} } < \infty$

Then there is at most one measure $\rho$ on the product space $\struct {X \times Y, \Sigma_1 \otimes \Sigma_2}$ such that:

$\forall G_1 \in \GG_1, G_2 \in \GG_2: \map \rho {G_1 \times G_2} = \map \mu {G_1} \, \map \nu {G_2}$

That is, there can be at most one product measure on $\struct {X \times Y, \Sigma_1 \otimes \Sigma_2}$.