Measure, Integration & Real Analysis · Problem 9B.4

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · singular-measures-total-variation-norm

Problem

Suppose ν\nu and μ\mu are complex measures on a measurable space (X,S)(X, \mathcal{S}). Prove that if ν⊥μ\nu \perp \mu, then ∣ν+μ∣=∣ν∣+∣μ∣|\nu + \mu| = |\nu| + |\mu| and ∥ν+μ∥=∥ν∥+∥μ∥\|\nu + \mu\| = \|\nu\| + \|\mu\|.

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