\begin{definition}[Definition:Jordan Decomposition of Complex Measure]
Let $\struct {X, \Sigma}$ be a measurable space. 
Let $\mu$ be a complex measure on $\struct {X, \Sigma}$ with real part $\mu_R$ and imaginary part $\mu_I$. 
Let $\tuple {\mu_1, \mu_2}$ be the Jordan decomposition of $\mu_R$.
Let $\tuple {\mu_3, \mu_4}$ be the Jordan decomposition of $\mu_I$.
Then: 
:$\mu = \mu_1 - \mu_2 + i \paren {\mu_3 - \mu_4}$
and we say that $\tuple {\mu_1, \mu_2, \mu_3, \mu_4}$ is the '''Jordan decomposition''' of $\mu$.
\end{definition}