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

Linear Maps on Hilbert Spaces — Sheldon Axler

Mathematics — Measure Theory · hahn-decomposition-uniqueness

Problem

Suppose ν\nu is a real measure on a measurable space (X,S)(X, \mathcal{S}). Prove that the Hahn decomposition of ν\nu is almost unique, in the sense that if A,BA, B and A′,B′A', B' are pairs satisfying the Hahn Decomposition Theorem (9.23), then ∣ν∣(A∖A′)=∣ν∣(A′∖A)=∣ν∣(B∖B′)=∣ν∣(B′∖B)=0.|\nu|(A \setminus A') = |\nu|(A' \setminus A) = |\nu|(B \setminus B') = |\nu|(B' \setminus B) = 0.

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