Measure, Integration & Real Analysis · Problem 9A.7

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · separability-banach-space-measures

Problem

(a) Suppose

\mathcal{B}

is the collection of Borel subsets of

\mathbf{R}

. Show that the Banach space

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

is not separable. (b) Give an example of a measurable space

(X, \mathcal{S})

such that the Banach space

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

is infinite-dimensional and separable.