\begin{definition}[Definition:Lower Integral]
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:
:$\ds \underline {\int_a^b} \map f x \rd x = \sup_P \map L P$
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 $P$.
\end{definition}