Measure, Integration & Real Analysis · Problem 5A.10

Product Measures — Sheldon Axler

Mathematics — Measure Theory · product-measure-uniqueness

Problem

Suppose (X,S,μ)(X, \mathcal{S}, \mu) and (Y,T,ν)(Y, \mathcal{T}, \nu) are σ\sigma-finite measure spaces. Prove that if ω\omega is a measure on S⊗T\mathcal{S} \otimes \mathcal{T} such that ω(A×B)=μ(A)ν(B)\omega(A \times B) = \mu(A)\nu(B) for all A∈SA \in \mathcal{S} and all B∈TB \in \mathcal{T}, then ω=μ×ν\omega = \mu \times \nu. [The exercise above means that μ×ν\mu \times \nu is the unique measure on S⊗T\mathcal{S} \otimes \mathcal{T} that behaves as we expect on measurable rectangles.]

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