\begin{definition}[Definition:Jordan Decomposition]
Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\tuple {P, N}$ be a Hahn decomposition of $\mu$. 
Define:
:$\map {\mu^+} A = \map \mu {A \cap P}$
and:
:$\map {\mu^-} A = -\map \mu {A \cap N}$
for each $A \in \Sigma$. 
Then from the Jordan Decomposition Theorem, we have: 
:$\mu = \mu^+ - \mu^-$
and we say that $\tuple {\mu^+, \mu^-}$ is the '''Jordan decomposition''' corresponding to $\tuple {P, N}$.
\end{definition}