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

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · total-variation-supremum-formula

Problem

Suppose ν\nu is a complex measure on a measurable space (X,S)(X, \mathcal{S}). Prove that if E∈SE \in \mathcal{S} then ∣ν∣(E)=sup⁡{∑k=1∞∣ν(Ek)∣:E1,E2,… is a disjoint sequence in S such that E=⋃k=1∞Ek}.|\nu|(E) = \sup\left\{\sum_{k=1}^{\infty} |\nu(E_k)| : E_1, E_2, \ldots \text{ is a disjoint sequence in } \mathcal{S} \text{ such that } E = \bigcup_{k=1}^{\infty} E_k\right\}.

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