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Hahn decomposition theorem (Theorem)

Let $\mu$ be a signed measure in the measurable space $(\Omega,\mathscr{S})$ There are two measurable sets $A$ and $B$ such that:

  1. $A\cup B = \Omega$ and $A\cap B = \emptyset$
  2. $\mu(E)\geq 0$ for each $E\in\mathscr{S}$ such that $E\subset A$
  3. $\mu(E)\leq 0$ for each $E\in\mathscr{S}$ such that $E\subset B$

The pair $(A,B)$ is called a Hahn decomposition for $\mu$ This decomposition is not unique, but any other such decomposition $(A',B')$ satisfies $\mu(A'\vartriangle A) = \mu(B\vartriangle B') = 0$ (where $\vartriangle$ denotes the symmetric difference), so the two decompositions differ in a set of measure 0.




"Hahn decomposition theorem" is owned by Koro.
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Also defines:  Hahn decomposition
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Cross-references: measure, symmetric difference, measurable sets, measurable space, signed measure
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This is version 7 of Hahn decomposition theorem, born on 2003-02-10, modified 2008-05-01.
Object id is 4014, canonical name is HahnDecompositionTheorem.
Accessed 5810 times total.

Classification:
AMS MSC28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)

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