Definition:Integral of Positive Measurable Function over Measurable Set

Definition
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $A \in \Sigma$.

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

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


 * $\ds \int_A f \rd \mu = \int \paren {\chi_A \cdot f} \rd \mu$

where:


 * $\chi_A$ is the characteristic function of $A$
 * $\chi_A \cdot f$ is the pointwise product of $\chi_A$ and $f$
 * the integral sign on the denotes $\mu$-integration of the function $\chi_A \cdot f$.

Also see

 * Integral of Positive Measurable Function over Measurable Set is Well-Defined