# Definition:Lower Integral

## Definition

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f: \left[{a \,.\,.\, b}\right] \to \R$ be a bounded real function.

The lower integral of $f$ over $\left[{a \,.\,.\, b}\right]$ is defined as:

$\displaystyle \underline{\int_a^b} f \left({x}\right) \ \mathrm d x = \sup_P L \left({P}\right)$

where:

the supremum is taken over all subdivisions $P$ of $\left[{a \,.\,.\, b}\right]$
$L \left({P}\right)$ denotes the lower sum of $f$ on $\left[{a \,.\,.\, b}\right]$ belonging to $P$.