Definition:Integral of Measure-Integrable Function over Measurable Set

Definition

Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $E \in \Sigma$.

Let $f: X \to \overline \R$ be a $\mu$-integrable function.

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

$\ds \int_E f \rd \mu := \int \chi_E \cdot f \rd \mu$

where:

$\chi_E$ is the characteristic function of $E$

$\chi_E \cdot f$ is the pointwise product of $\chi_E$ and $f$

the integral sign on the right hand side denotes $\mu$-integration of the function $\chi_E \cdot f$.

Also see

• Integral of Integrable Function over Measurable Set is Well-Defined

• Results about integrals of measure-integrable functions can be found here.

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $10.7$