Measure, Integration & Real Analysis · Problem 9B.14

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · absolute-continuity-epsilon-delta

Problem

Suppose μ\mu is a (positive) σ\sigma-finite measure on a measurable space (X,S)(X, \mathcal{S}) and ν\nu is a complex measure on (X,S)(X, \mathcal{S}). Show that the following are equivalent. (a) ν≪μ\nu \ll \mu. (b) For every ε>0\varepsilon > 0, there exists δ>0\delta > 0 such that ∣ν(E)∣<ε|\nu(E)| < \varepsilon for every set E∈SE \in \mathcal{S} with μ(E)<δ\mu(E) < \delta. (c) For every ε>0\varepsilon > 0, there exists δ>0\delta > 0 such that ∣ν∣(E)<ε|\nu|(E) < \varepsilon for every set E∈SE \in \mathcal{S} with μ(E)<δ\mu(E) < \delta.

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