Articles containing proofs | Theorems in measure theory
In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space and any signed measure defined on the -algebra , there exist two -measurable sets, and , of such that: 1. * and . 2. * For every such that , one has , i.e., is a positive set for . 3. * For every such that , one has , i.e., is a negative set for . Moreover, this decomposition is essentially unique, meaning that for any other pair of -measurable subsets of fulfilling the three conditions above, the symmetric differences and are -null sets in the strong sense that every -measurable subset of them has zero measure. The pair is then called a Hahn decomposition of the signed measure . (Wikipedia).
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