Lower and upper Darboux sums over a grid partition in $\mathbb{R}^m$

Let $f:Q\to\mathbb R$ be bounded on a nondegenerate rectangle and let $P$ be a grid. For each cell $Q_i$, put $m_i:=\inf f[Q_i],\quad M_i:=\sup f[Q_i],\quad L(f,P):=\sum_i m_i\operatorname{vol}(Q_i),\quad U(f,P):=\sum_i M_i\operatorname{vol}(Q_i).$ The extrema exist as finite reals because each nonempty image is bounded (Lower bound, bounded below, bounded set, Complete ordered field (least-upper-bound property), Every nonempty set bounded below has an infimum, Greatest lower bound (infimum), Suprema and infima are unique), and the sums use the iterated convention of Grid partitions of a rectangle in $\mathbb{R}^m$, their cells, refinements and mesh.

Since $m_i\le M_i$ and cell volumes are nonnegative, $L(f,P)\le U(f,P)$. Moreover $U(f,P)-L(f,P)=\sum_i(M_i-m_i)\operatorname{vol}(Q_i),$ the sum of cell oscillations weighted by volume (Laws of finite sums and finite products).