Hahn decomposition theorem
Let $\mu $ be a signed measure in the measurable space^{} $(\mathrm{\Omega},\mathcal{S})$. There are two measurable sets $A$ and $B$ such that:

1.
$A\cup B=\mathrm{\Omega}$ and $A\cap B=\mathrm{\varnothing}$;

2.
$\mu (E)\ge 0$ for each $E\in \mathcal{S}$ such that $E\subset A$;

3.
$\mu (E)\le 0$ for each $E\in \mathcal{S}$ 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 $({A}^{\prime},{B}^{\prime})$ satisfies $\mu ({A}^{\prime}\mathrm{\u25b3}A)=\mu (B\mathrm{\u25b3}{B}^{\prime})=0$ (where $\mathrm{\u25b3}$ denotes the symmetric difference^{}), so the two decompositions differ in a set of measure^{} 0.
Title  Hahn decomposition theorem 

Canonical name  HahnDecompositionTheorem 
Date of creation  20130322 13:26:59 
Last modified on  20130322 13:26:59 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  10 
Author  Koro (127) 
Entry type  Theorem 
Classification  msc 28A12 
Defines  Hahn decomposition 