Definition:Upper Sum

Definition
Let $\left[{a \,. \, . \, b}\right]$ be a closed interval of the set $\R$ of real numbers.

Let $P = \left\{{x_0, x_1, x_2, \ldots, x_{n-1}, x_n}\right\}$ be a subdivision of $\left[{a \,. \, . \, b}\right]$.

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

For all $\nu \in 1, 2, \ldots, n$, let $\left[{x_{\nu - 1} \,. \, . \, x_{\nu}}\right]$ be a closed subinterval of $\left[{a \,. \, . \, b}\right]$.

Let $M_\nu^{\left({f}\right)}$ be the supremum of $f \left({x}\right)$ on the interval $\left[{x_{\nu - 1} \,. \, . \, x_{\nu}}\right]$.

Then:
 * $\displaystyle U^{\left({f}\right)} \left({P}\right) = \sum_{\nu=1}^n M_\nu^{\left({f}\right)} \left({x_{\nu} - x_{\nu - 1}}\right)$

is called the '''upper sum of $f \left({x}\right)$ on $\left[{a \,. \, . \, b}\right]$ belonging to the subdivision $P$'''.

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

Compare Lower Sum.