Measure, Integration & Real Analysis · Problem 9B.11

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · absolute-continuity-real-measure

Problem

Suppose

\mu

is a (positive) measure on a measurable space

(X, \mathcal{S})

and

\nu

is a real measure on

(X, \mathcal{S})

. Show that

\nu \ll \mu

if and only if

\nu^+ \ll \mu

and

\nu^- \ll \mu