Definition:Integral of Integrable Function

Definition
Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $f: X \to \overline{\R}$, $f \in \mathcal{L}^1 \left({\mu}\right)$ be a $\mu$-integrable function.

Then the $\mu$-integral of $f$, $\displaystyle \int f \, \mathrm d\mu$, is defined as:


 * $\displaystyle \int f \, \mathrm d\mu := \int f^+ \, \mathrm d\mu - \int f^- \, \mathrm d\mu$

where $f^+$ and $f^-$ are the positive and negative parts of $f$, respectively.

Also known as
The $\mu$-integral is also sometimes called the (abstract) Lebesgue integral.

Also see

 * Integrable Function