Measure, Integration & Real Analysis · Problem 9A.2

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · total-variation-absolute-value

Problem

Suppose ν\nu is a real measure on (X,S)(X, \mathcal{S}). Define μ:S→[0,∞)\mu: \mathcal{S} \to [0, \infty) by μ(E)=∣ν(E)∣.\mu(E) = |\nu(E)|. Prove that μ\mu is a (positive) measure on (X,S)(X, \mathcal{S}) if and only if the range of ν\nu is contained in [0,∞)[0, \infty) or the range of ν\nu is contained in (−∞,0](-\infty, 0].

Related theory

Browse the problem library