Hahn decomposition theorem

Let $\mu$ be a signed measure in the measurable space $(\mathrm{\Omega },𝒮)$. 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 $E\in 𝒮$ such that $E\subset A$;

3. $\mu (E)\le 0$ for each $E\in 𝒮$ 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 $\mu (A^{\prime }\mathrm{△}A)=\mu (B\mathrm{△}{B}^{\prime })=0$ (where $\mathrm{△}$ 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	2013-03-22 13:26:59
Last modified on	2013-03-22 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