Definition:Integral of Bounded Measurable Function with respect to Finite Signed 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 finite signed measure on $\struct {X, \Sigma}$.

Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.

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

$\ds \int f \rd \mu = \int f \rd \mu^+ - \int f \rd \mu^-$

Also see

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

Sources

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