Definition:Upper Integral

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Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f: \left[{a \,.\,.\, b}\right] \to \R$ be a bounded real function.

The upper integral of $f$ over $\left[{a \,.\,.\, b}\right]$ is defined as:

$\displaystyle \overline{\int_a^b} f \left({x}\right) \, \mathrm d x = \inf_P U \left({P}\right)$

where the infimum is taken over all subdivisions $P$ of $\left[{a \,.\,.\, b}\right]$, and $U \left({P}\right)$ denotes the upper sum of $f$ on $\left[{a \,.\,.\, b}\right]$ belonging to $P$.

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