\begin{definition}[Definition:Real Part of Complex Measure]
Let $\struct {X, \Sigma}$ be a measurable space. 
Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.
From Decomposition of Complex Measure into Finite Signed Measures, there exists unique finite measures $\mu_R$ and $\mu_I$ such that: 
:$\mu = \mu_R + i \mu_I$
We say that $\mu_R$ is the '''real part''' of $\mu$.
\end{definition}