Measure, Integration & Real Analysis · Problem 9B.3

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · singular-measures-equivalences

Problem

Suppose ν\nu and μ\mu are complex measures on a measurable space (X,S)(X, \mathcal{S}). Show that the following are equivalent. (a) ν⊥μ\nu \perp \mu. (b) ∣ν∣⊥∣μ∣|\nu| \perp |\mu|. (c) Re⁡ν⊥μ\operatorname{Re}\nu \perp \mu and Im⁡ν⊥μ\operatorname{Im}\nu \perp \mu.

Browse the problem library