= Solution
The <product measure> $\mu\otimes\nu$ is the unique <probability measure> on the <sigma-algebra> generated by measurable rectangles for which
$$
\boxed{(\mu\otimes\nu)(C\times D)=\mu(C)\nu(D)
\quad\text{for all }C,D\in\mathcal B(\mathbb R).}
$$
Here the printed product notation $\mathcal B(\mathbb R)\times\mathcal B(\mathbb R)$ denotes the generated <product sigma-algebra>, conventionally written $\mathcal B(\mathbb R)\otimes\mathcal B(\mathbb R)$; it is not just the collection of rectangles. On $\mathbb R^2$ it equals $\mathcal B(\mathbb R^2)$: coordinate projections show that rectangles are <Borel sets>, while open rectangles with rational endpoints give every open <set> as a <countable union> and hence generate the <Borel sigma-algebra>.
Existence is the <product measure> theorem for finite <measures>, which are in particular <sigma-finite measures>. Uniqueness follows because the rectangles form a <pi-system> generating the <product sigma-algebra>, and two finite <measures> agreeing there agree everywhere by the <Dynkin lemma>.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[§1.7, Theorem 1.7.1, printed p. 38: existence and characterization on rectangles]{ref}
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