Definition:Lower Sum

Definition
Let $\closedint a b$ be a closed real interval.

Let $f: \closedint a b \to \R$ be a bounded real function.

Let $P = \set {x_0, x_1, x_2, \ldots, x_n}$ be a finite subdivision of $\closedint a b$.

For all $\nu \in \set {1, 2, \ldots, n}$, let $m_\nu^{\paren f}$ be the infimum of $f$ on the interval $\closedint {x_{\nu - 1} } {x_\nu}$.

Then:
 * $\ds \map {L^{\paren f} } P = \sum_{\nu \mathop = 1}^n m_\nu^{\paren f} \paren {x_\nu - x_{\nu - 1} }$

is called the lower sum of $f$ on $\closedint a b$ belonging (or with respect) to (the subdivision) $P$.

If there is no ambiguity as to what function is under discussion, $m_\nu$ and $\map L P$ are usually used.

Also known as
The notation $\map L {f, P}$ or $\map L {P, f}$ can be used in place of $\map {L^{\paren f} } P$.

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

Also see

 * Definition:Upper Sum
 * Definition:Upper Integral
 * Definition:Lower Integral


 * Definition:Riemann Sum