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.