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

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · absolute-continuity-condition-on-h

Problem

Suppose

\mu

is a (positive) finite measure on a measurable space

(X, \mathcal{S})

and

h

is a nonnegative function in

\mathcal{L}^1(\mu)

. Thus

h\, d\mu \ll d\mu

. Find a reasonable condition on

h

that is equivalent to the condition

d\mu \ll h\, d\mu