Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 26K / a

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Solution

The product sigma-algebra is

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

The product measure

μ_{1} \otimes μ_{2}

is the probability measure on this sigma-algebra satisfying

(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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