\begin{definition}[Definition:Integral of Integrable Function over Measurable Set]
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 {{RHS}} denotes $\mu$-integration of the function $\chi_E \cdot f$.
\end{definition}