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 $\left[{a \,. \, . \, b}\right]$, and $L \left({P}\right)$ denotes the lower sum of $f$ on $\left[{a \,. \, . \, b}\right]$ belonging to $P$.

Also see

 * Upper Integral
 * Definite Integral