Measure, Integration & Real Analysis · Problem 9B.6

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · singular-measures-closed-subspace

Problem

Suppose μ\mu is a complex or positive measure on a measurable space (X,S)(X, \mathcal{S}). Prove that {ν∈MF(S):ν⊥μ}\{\nu \in \mathcal{M}_{\mathbf{F}}(\mathcal{S}) : \nu \perp \mu\} is a closed subspace of MF(S)\mathcal{M}_{\mathbf{F}}(\mathcal{S}).

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