Measure zero and content zero in $\mathbb{R}^m$ by countable and finite cube covers

Fix $m\ge1$. A closed cube is a rectangle $\prod_{j<m}[a_j,a_j+\ell]$ with $\ell\ge0$; its volume is $\ell^m$. A set $E\subseteq\mathbb R^m$ is null when, for every $\varepsilon>0$, it is covered by a sequence of closed cubes whose nonnegative volume series converges with sum at most $\varepsilon$. It has content zero when such a cover can be finite.

The series and finite sums are Series, partial sums, convergence and the sum, divergence, and the tail series and Finite sums and finite products, by recursion, and their nonnegative bounds use A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum and Laws of finite sums and finite products. Both properties pass to subsets. Padding a finite cover with degenerate zero-volume cubes proves that content zero implies null. This terminology defines only cover-nullity; it does not define a measure on arbitrary sets.