User:Caliburn/s/mt/Definition:Integral of Bounded Measurable Function with respect to Complex Measure

Definition

Let $\struct {X, \Sigma}$ be a measurable space.

Let $f : X \to \R$ be a bounded $\Sigma$-measurable function.

Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.

Let $\mu_R$ be the real part of $\mu$.

Let $\mu_I$ be the imaginary part of $\mu$.

Then the $\mu$-integral of $f$ is defined by:

$\ds \int f \rd \mu = \int f \rd \mu_R + i \int f \rd \mu_I$

where the integral sign on the right hand side denotes integration with respect to the signed measures $\mu_R$ and $\mu_I$.

Also see

• Integral of Bounded Measurable Function with respect to Complex Measure is Well-Defined

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.2$: Absolute Continuity: