Measure, Integration & Real Analysis · Problem 9B.2

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · singular-measures-characterization

Problem

Suppose μ\mu is a (positive) measure and g,h∈L1(μ)g, h \in \mathcal{L}^1(\mu). Prove that g dμ⊥h dμg\, d\mu \perp h\, d\mu if and only if g(x)h(x)=0g(x)h(x) = 0 for almost every x∈Xx \in X.

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