 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-2/26k/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

A_{1} \otimes A_{2} = σ {A_{1} \times A_{2} : A_{i} \in A_{i}} .

μ_{1} \otimes μ_{2}

(μ_{1} \otimes μ_{2}) (A_{1} \times A_{2}) = μ_{1} (A_{1}) μ_{2} (A_{2}) .

The measurable rectangles form a pi-system generating

A_{1} \otimes 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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