Category:Measure-Integrable Functions

This category contains results about Measure-Integrable Functions.
Definitions specific to this category can be found in Definitions/Measure-Integrable Functions.

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $\MM_{\overline \R}$ denote the space of $\Sigma$-measurable, extended real-valued functions .

Let $f \in \MM_{\overline \R}, f: X \to \overline \R$ be a measurable function.

Then $f$ is said to be $\mu$-integrable if and only if:

$\ds \int f^+ \rd \mu < +\infty$

and

$\ds \int f^- \rd \mu < +\infty$

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

The integral signs denote $\mu$-integration of positive measurable functions.