Definition:Upper Sum

Let $$\left[{a \,. \, . \, b}\right]$$ be a closed interval of the set $$\mathbb{R}$$ 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 $$f: \mathbb{R} \to \mathbb{R}$$ 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 $$U^{\left({f}\right)} \left({P}\right) = \sum_{\nu=1}^n M_\nu^{\left({f}\right)} \left({x_{\nu} - x_{\nu - 1}}\right)$$ 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.