Hahn?Jordan decomposition

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

Related Topics:
Mathematics - Measure - Positive

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A signed measure μ on a sigma-algebra Σ is a countably additive function which takes values in the reals extended to pm infty, or in other words in the interval . 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

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mu_{+} = mu(X igcap Omega_{+}) and

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mu_{-} = -mu(X igcap Omega_{-})

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for every set X in Σ.

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