Definition:Upper 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 supremum of $f$ on the interval $\closedint {x_{\nu - 1} } {x_\nu}$.

Then:

$\ds \map {U^{\paren f} } P = \sum_{\nu \mathop = 1}^n M_\nu^{\paren f} \paren {x_\nu - x_{\nu - 1} }$

is called the upper 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 $\map U P$ are often seen.

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

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