Definition:Upper 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 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 sum of $f$ on $\closedint a b$ belonging to $P$.

Also see

 * Lower Integral
 * Definite Integral