Definition:Real Part of Complex Measure

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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 $\mu$.


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