Conventions and proved scope for the Riemann integral in $\mathbb{R}^m$ and Jordan content

Throughout, $m\ge1$. Rectangles and grids are axis-parallel. The multidimensional Darboux and tagged integrals are defined on nondegenerate rectangles, and integration over a Jordan set chooses a nondegenerate bounding rectangle. Degenerate rectangles still have geometric volume and Jordan content $0$, but no competing integral convention is introduced for them. Nullity in Measure zero and content zero in $\mathbb{R}^m$ by countable and finite cube covers uses cube covers, while Jordan outer content in Jordan inner and outer content and Jordan measurable bounded sets in $\mathbb{R}^m$ uses arbitrary finite rectangle covers. The one-dimensional dictionaries are At $m=1$, nondegenerate multidimensional rectangles, grid sums and the integral are exactly the published one-dimensional notions and At $m=1$, cube-nullity and cube-content-zero are exactly the published interval-cover notions.

The historical Lebesgue criterion Lebesgue's criterion in $\mathbb{R}^m$: a bounded function on a closed nondegenerate rectangle is Riemann integrable iff its discontinuity set is null uses only cover-nullity and no Lebesgue measure or integral. The proved image results are the equal-dimensional Lipschitz theorem A Lipschitz map $\mathbb{R}^m\to\mathbb{R}^m$ sends null sets to null sets and the graph theorem The graph of a continuous function on a closed nondegenerate rectangle in $\mathbb{R}^m$ has content zero in $\mathbb{R}^{m+1}$. No general continuously differentiable image theorem is asserted.

Jordan measurability is related to null boundaries by A bounded set in $\mathbb{R}^m$ is Jordan measurable iff its boundary is null, equivalently of content zero. Integration over a Jordan set uses the zero-extension convention of The Riemann integral of a bounded function over a bounded Jordan measurable set; no integration over arbitrary bounded sets is defined here.