Uniqueness of Product Measures

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Theorem

Let $\left({X, \Sigma_1, \mu}\right)$ and $\left({Y, \Sigma_2, \nu}\right)$ be measure spaces.

Let $\mathcal G_1$ and $\mathcal G_2$ be generators for $\Sigma_1$ and $\Sigma_2$, respectively.


Suppose that $\mathcal G_1$ and $\mathcal G_2$ are closed under intersection.

Suppose further that there are exhausting sequences $\left({G_{1,n}}\right)_{n \in \N}$ and $\left({G_{2,n}}\right)_{n \in \N}$ in $\mathcal G_1$ and $\mathcal G_2$, respectively, such that:

$\forall n \in \N: \mu \left({G_{1,n}}\right) < \infty$
$\forall n \in \N: \nu \left({G_{2,n}}\right) < \infty$


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

$\forall G_1 \in \mathcal G_1, G_2 \in \mathcal G_2: \rho \left({G_1 \times G_2}\right) = \mu \left({G_1}\right) \nu \left({G_2}\right)$

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


Proof


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