Measure, Integration & Real Analysis · Problem 9B.5

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · singular-positive-measures-norm-condition

Problem

Suppose ν\nu and μ\mu are finite (positive) measures on a measurable space (X,S)(X, \mathcal{S}). Prove that ν⊥μ\nu \perp \mu if and only if ∥ν−μ∥=∥ν∥+∥μ∥\|\nu - \mu\| = \|\nu\| + \|\mu\|.

Browse the problem library