Measure, Integration & Real Analysis · Problem 9B.10

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · absolute-continuity-equivalences

Problem

Suppose μ\mu is a (positive) 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) ∣ν∣≪μ|\nu| \ll \mu. (c) Re⁡ν≪μ\operatorname{Re}\nu \ll \mu and Im⁡ν≪μ\operatorname{Im}\nu \ll \mu.
Intuitive explanation

The key insight is that absolute continuity is really about when sets of measure zero stay "invisible" to the measure in question. Since a complex measure ν\nu can be decomposed into its real and imaginary parts, and its total variation ∣ν∣|\nu| bounds all of them, you need to show these different ways of capturing ν\nu's "size" all have the same null sets as μ\mu. The chain of reasoning flows through how the variation of ν\nu relates to variations of its components.

Intuitive explanation

The key insight is that absolute continuity is really about when sets of measure zero stay "invisible" to the measure in question. Since a complex measure ν\nu can be decomposed into its real and imaginary parts, and its total variation ∣ν∣|\nu| bounds all of them, you need to show these different ways of capturing ν\nu's "size" all have the same null sets as μ\mu. The chain of reasoning flows through how the variation of ν\nu relates to variations of its components.

Related theory

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