Measure, Integration & Real Analysis · Problem 9A.1

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · total-variation-real-measures

Problem

Prove or give a counterexample: If

\nu

is a real measure on a measurable space

(X, \mathcal{S})

and

A, B \in \mathcal{S}

are such that

\nu(A) \geq 0

and

\nu(B) \geq 0

, then

\nu(A \cup B) \geq 0