\begin{definition}[Definition:Hahn Decomposition]
Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
We say that $\tuple {P, N}$ is a '''Hahn decomposition''' of $\mu$ {{iff}}:
:$(1): \quad$ $P$ and $N$ are disjoint.
:$(2): \quad$ $X = P \cup N$
:$(3): \quad$ $P$ is $\mu$-positive and $N$ is $\mu$-negative.
\end{definition}