Definition:Upper Integral/Rectangle

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

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

The upper integral of $f$ over $R$ is defined as:
 * $\ds \overline{\int_R} \map f x \rd x = \inf_P \map U P$

where:
 * $P$ ranges over all finite subdivisions of $R$.
 * $\map U P$ denotes the upper sum of $f$ on $R$ with respect to $P$.