Let $f: Q\to \mathbb R$ be a bounded function defined on the rectangle $Q$ and let $P$ be a partition of a set of $Q$. For each subrectangle $R$ determined by $P$, let $m_R(f) = \inf\limits_{x\in R} f(x)$. The lower sum for $f$ determined by $P$ is \(L(f,P) = \sum_{R} m_R(f)\text{vol}(R)\) where $\text{vol}$ is the volume of the rectangle $R$.