Articles containing proofs | Theorems in measure theory

Hahn decomposition theorem

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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Related pages

Series (mathematics) | Positive and negative sets | Signed measure | Mathematics | Q.E.D. | Symmetric difference | Null set | Universal property | Mathematical induction