Definition:Lower Integral

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Definition

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

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


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

$\displaystyle \underline {\int_a^b} \map f x \rd x = \sup_P \map L P$

where:

the supremum is taken over all subdivisions $P$ of $\closedint a b$
$\map L P$ denotes the lower sum of $f$ on $\closedint a b$ belonging to $P$.


Also see


Sources