Definition:Lower 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 lower integral of $f$ over $R$ is defined as:
 * $\ds \underline{\int_R} \map f x \rd x = \sup_P \map L P$

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