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

Linear Maps on Hilbert Spaces — Sheldon Axler

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

Problem

Suppose

(X, \mathcal{S}, \mu)

is a (positive) measure space. Prove that

\{h\, d\mu : h \in \mathcal{L}^1(\mu)\}

is a closed subspace of

\mathcal{M}_{\mathbf{F}}(\mathcal{S})