Characterization of Integrable Functions

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

Let $f: X \to \overline{\R}, f \in \mathcal{M}_{\overline{\R}}$ be a $\Sigma$-measurable function.

Then the following are equivalent:


 * $(1): \quad f \in \mathcal{L}_{\overline{\R}} \left({\mu}\right)$, i.e., $f$ is $\mu$-integrable
 * $(2): \quad$ The positive and negative parts $f^+$ and $f^-$ are $\mu$-integrable
 * $(3): \quad$ The absolute value $\left\vert{f}\right\vert$ of $f$ is $\mu$-integrable
 * $(4): \quad$ There exists an $\mu$-integrable function $g: X \to \overline{\R}$ such that $\left\vert{f}\right\vert \le g$ pointwise