Definition:Integral of Integrable Function over Measurable Set

Definition
Let $\left({X, \Sigma, \mu}\right)$ 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:


 * $\displaystyle \int_E f \ \mathrm d \mu := \int \chi_E \cdot f \ \mathrm d \mu$

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 RHS denotes $\mu$-integration of the function $\chi_E \cdot f$.