Definition:Upper Sum/Rectangle

Definition
Let $R$ be a closed rectangle in $\R^n$.

Let $f : R \to \R$ be a bounded real-valued function.

Let $P$ be a finite subdivision of $R$.

Let $S$ be the set of subrectangles of $P$.

For every:
 * $r = \closedint {a_1} {b_1} \times \dotso \times \closedint {a_n} {b_n} \in S$

define:
 * $\ds M_r^{\paren f} = \sup_{x \mathop \in r} \map f x$
 * $\ds \map v r = \prod_{1 \mathop \le i \mathop \le n} \paren {b_i - a_i}$

Then:
 * $\ds \map {U^{\paren f}} P = \sum_{r \mathop \in S} M_r^{\paren f} \map v r$

is called the upper sum of $f$ on $R$ with respect to $P$.

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