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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 4316 times total.

Classification:
AMS MSC: 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities)
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