Definition:Lower Integral

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

 * Definition:Upper Integral
 * Definition:Definite Integral