Hahn-Jordan decomposition

In mathematics, the Hahn-Jordan decomposition breaks a signed measure into two parts, a positive and a negative part.

A signed measure μ on a sigma-algebra Σ is a countably additive function which takes values in the reals extended to <math>\pm \infty<math>, or in other words in the interval <math>[-\infty, \infty]<math>. The Hahn-Jordan decomposition tells us that the measure space Ω can be partitioned into two disjoint sets contained in Σ, Ω⁺ and Ω^-, such that μ is nonnegative for every set contained in Ω⁺ and nonpositive for every set contained in Ω^-. Consequently μ is broken up into two ordinary measures μ₊ and μ_-, such that μ = μ₊ – μ_-, by taking <math>\mu_{+} = \mu(X \bigcap \Omega_{+})<math> and <math>\mu_{-} = -\mu(X \bigcap \Omega_{-})<math> for every set X in Σ.