# Hahn decomposition 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. 1.

$A\cup B=\Omega$ and $A\cap B=\emptyset$;

2. 2.

$\mu(E)\geq 0$ for each $E\in\mathscr{S}$ such that $E\subset A$;

3. 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 $(A^{\prime},B^{\prime})$ satisfies $\mu(A^{\prime}\vartriangle A)=\mu(B\vartriangle B^{\prime})=0$ (where $\vartriangle$ denotes the symmetric difference), so the two decompositions differ in a set of measure 0.

Title Hahn decomposition theorem HahnDecompositionTheorem 2013-03-22 13:26:59 2013-03-22 13:26:59 Koro (127) Koro (127) 10 Koro (127) Theorem msc 28A12 Hahn decomposition