Measure, Integration & Real Analysis · Problem 5B.3

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · region-under-graph-measurability

Problem

Suppose (X,S)(X, \mathcal{S}) is a measurable space and f:X→[0,∞]f : X \to [0, \infty] is a function. Let B\mathcal{B} denote the σ\sigma-algebra of Borel subsets of (0,∞)(0, \infty). Prove that Uf∈S⊗BU_f \in \mathcal{S} \otimes \mathcal{B} if and only if ff is an S\mathcal{S}-measurable function.

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