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

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 $P$.

Also see

 * Definition:Upper Integral
 * Definition:Definite Integral