MathGloss

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 $M_R(f) = \sup\limits_{x\in R} f(x)$. The lower sum for $f$ determined by $P$ is $$U(f,P) = \sum_{R} M_R(f)\text{vol}(R)$$ where $\text{vol}$ is the volume of the rectangle $R$.

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