Definition:Lower Darboux 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 Darboux sum of $f$ on $R$ with respect to $P$.

Sources

• 1965: Michael Spivak: Calculus on Manifolds: $3$ Integration