|
|
|
|
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:
- $A\cup B = \Omega$ and $A\cap B = \emptyset$
- $\mu(E)\geq 0$ for each $E\in\mathscr{S}$ such that $E\subset A$
- $\mu(E)\leq 0$ 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.
|
|
(view preamble | get metadata)
| Also defines: |
Hahn decomposition |
|
|
Cross-references: measure, symmetric difference, measurable sets, measurable space, signed measure
There is 1 reference to this entry.
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 MSC: | 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
