Definition:Imaginary 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 $\mu_I$ is the imaginary part of $\mu$.

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.1$: Signed and Complex Measures