Definition:Jordan Decomposition of Complex Measure

Definition

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 $\mu$.

Sources

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