Definition:Real Part of Complex Measure

Definition
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 $\mu$.