Given a Hahn decomposition X = P ⊔ N, set ν⁺(A) = ν(A∩P), ν⁻(A) = −ν(A∩N). Then ν⁺, ν⁻ are mutually singular finite positive measures with ν = ν⁺ − ν⁻, and the total variation |ν| = ν⁺ + ν⁻ is the minimal positive measure dominating ν.
Given a Hahn decomposition X = P ⊔ N, set ν⁺(A) = ν(A∩P), ν⁻(A) = −ν(A∩N). Then ν⁺, ν⁻ are mutually singular finite positive measures with ν = ν⁺ − ν⁻, and the total variation |ν| = ν⁺ + ν⁻ is the minimal positive measure dominating ν.