Tagged grid partitions and Riemann sums in $\mathbb{R}^m$

A tagging of a grid $P$ assigns to every cell $Q_i$ a point $\xi_i\in Q_i$. The lower corner is a canonical tagging, so taggings exist without choice. The Riemann sum is $S(f,P,\xi):=\sum_i f(\xi_i)\operatorname{vol}(Q_i),$ with the iterated sum convention of Grid partitions of a rectangle in $\mathbb{R}^m$, their cells, refinements and mesh.

The tagged sums converge with mesh to $I$ if for every $\varepsilon>0$ some $\delta>0$ makes $|S(f,P,\xi)-I|<\varepsilon$ for every tagged grid with mesh below $\delta$. Finite cellwise selections used in proofs are licensed by Every natural-number-indexed list of nonempty sets has a choice function on its family of values and Choice function, not by countable choice.

For bounded $f$, each tag lies in its cell, so termwise inequalities and nonnegative volumes give $L(f,P)\le S(f,P,\xi)\le U(f,P)$ (Lower and upper Darboux sums over a grid partition in $\mathbb{R}^m$, Lower bound, bounded below, bounded set).