\begin{definition}[Definition:Integral of Integrable Function]
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f: X \to \overline \R$, $f \in \map {\LL^1} \mu$ be a $\mu$-integrable function.
Then the '''$\mu$-integral of $f$''' is defined by:
:$\ds \int f \rd \mu := \int f^+ \rd \mu - \int f^- \rd \mu$
where $f^+$ and $f^-$ are the positive and negative parts of $f$, respectively.
\end{definition}