# 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:

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

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 $P$.