= Solution
The <product sigma-algebra> is
$$
\mathcal A_1\otimes\mathcal A_2
=\sigma\{A_1\times A_2:A_i\in\mathcal A_i\}.
$$
The <product measure> $\mu_1\otimes\mu_2$ is the probability measure on this sigma-algebra satisfying
$$
(\mu_1\otimes\mu_2)(A_1\times A_2)=\mu_1(A_1)\mu_2(A_2).
$$
The measurable rectangles form a <pi-system> generating $\mathcal A_1\otimes\mathcal A_2$. Any two probability measures agreeing on them therefore agree on the generated sigma-algebra by the <pi-lambda theorem>, proving uniqueness.
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