\begin{definition}[Definition:Lower Sum]
Let $\closedint a b$ be a closed real interval.
Let $f: \closedint a b \to \R$ be a bounded real function.
Let $P = \set {x_0, x_1, x_2, \ldots, x_n}$ be a finite subdivision of $\closedint a b$.
For all $\nu \in \set {1, 2, \ldots, n}$, let $m_\nu^{\paren f}$ be the infimum of $f$ on the interval $\closedint {x_{\nu - 1} } {x_\nu}$.
Then:
:$\ds \map {L^{\paren f} } P = \sum_{\nu \mathop = 1}^n m_\nu^{\paren f} \paren {x_\nu - x_{\nu - 1} }$
is called the '''lower sum of $f$ on $\closedint a b$ belonging''' (or '''with respect''') '''to (the subdivision) $P$'''.
If there is no ambiguity as to what function is under discussion, $m_\nu$ and $\map L P$ are usually used.
\end{definition}