Definition:Upper Darboux Integral

Definition

Let $\closedint a b$ be a closed real interval.

Let $f: \closedint a b \to \R$ be a bounded real function.

The upper Darboux integral of $f$ over $\closedint a b$ is defined as:

$\ds \overline {\int_a^b} \map f x \rd x = \inf_P \map U P$

where:

the infimum is taken over all subdivisions $P$ of $\closedint a b$

$\map U P$ denotes the upper Darboux sum of $f$ on $\closedint a b$ belonging to $P$.

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

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

The upper Darboux 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 Darboux sum of $f$ on $R$ with respect to $P$.

The upper Darboux integral is also known just as the upper integral.

This entry was named for Jean-Gaston Darboux.