The lower and upper Darboux integrals over a nondegenerate rectangle in $\mathbb{R}^m$

For a bounded function $f:Q\to\mathbb R$ on a nondegenerate rectangle $Q\subseteq\mathbb R^m$, define $\underline{\int_Q}f:=\sup_P L(f,P),\qquad \overline{\int_Q}f:=\inf_P U(f,P),$ over all grid partitions $P$ of $Q$. The grid family is nonempty, since the endpoints in each coordinate give a one-cell grid. Every lower sum is at most every upper sum by a common refinement and Refinement raises multidimensional lower sums and lowers upper sums, with a quantitative boundary-slab estimate, so the two sets of sums are nonempty and bounded and the extrema exist (Complete ordered field (least-upper-bound property), Every nonempty set bounded below has an infimum, Greatest lower bound (infimum), Suprema and infima are unique).

The function is Riemann integrable over $Q$ when the two values agree. Their unique common real is $\int_Q f$. No integral is defined here for a degenerate rectangle, because the grid definition requires every coordinate interval to have distinct endpoints. This is the multidimensional Darboux definition; its agreement with the published one-dimensional definition is proved separately.