# Definition:Lower Sum

## Definition

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

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

Let $P = \left\{{x_0, x_1, x_2, \ldots, x_n}\right\}$ be a finite subdivision of $\left[{a \,.\,.\, b}\right]$.

For all $\nu \in \left\{{1, 2, \ldots, n}\right\}$, let $m_\nu^{\left({f}\right)}$ be the infimum of $f$ on the interval $\left[{x_{\nu - 1} \,.\,.\, x_{\nu}}\right]$.

Then:

$\displaystyle L^{\left({f}\right)} \left({P}\right) = \sum_{\nu \mathop = 1}^n m_\nu^{\left({f}\right)} \left({x_{\nu} - x_{\nu - 1}}\right)$

is called the lower sum of $f$ on $\left[{a \,.\,.\, b}\right]$ belonging (or with respect) to (the subdivision) $P$.

If there is no ambiguity as to what function is under discussion, $m_\nu$ and $L \left({P}\right)$ are often seen.

## Also known as

The notation $L \left({P, f}\right)$ can be used in place of $L^{\left({f}\right)} \left({P}\right)$.

The lower sum is also known as the lower Darboux sum or the lower Riemann sum.