The published refutations separating nullity from nowhere density hold verbatim for Lebesgue measure

Assume the Axiom of Countable Choice. Two notions of smallness for subsets of $\mathbb{R}$ are now in play: being ${\lambda }_1$-null, and being nowhere dense (Nowhere dense, meager (first category), residual, and second category subsets of $\mathbb{R}$). Neither implies the other, and the two published refutations transfer to Lebesgue measure without a new argument, because A subset of $\mathbb{R}$ has Lebesgue outer measure zero if and only if it has measure zero in the sense of countable closed-interval covers identifies ${\lambda }_1^{\ast }(A)=0$ with the covering condition of Measure zero (a countable cover by intervals of total length below every $\epsilon$) and content zero (a finite such cover) that those items are stated in.

Null does not imply nowhere dense. FALSE: every subset of $\mathbb{R}$ of measure zero is nowhere dense records the false claim and its witness. Read through the agreement theorem, the witness is a ${\lambda }_1$-null set whose closure is all of $\mathbb{R}$; the rationals of the line are one, and their nullity is also the case $n=1$ of Every at most countable subset of ${\mathbb{R}}^n$ is Lebesgue null; in particular ${\lambda }_1(\mathbb{Q})=0$.

Nowhere dense does not imply null. FALSE: every nowhere dense subset of $\mathbb{R}$ has measure zero records that false claim, and The Smith-Volterra-Cantor set is compact, perfect and nowhere dense, and does not have measure zero proves of the Smith–Volterra–Cantor set $S$ that it is compact, perfect and nowhere dense while no cover of it by intervals has total length below $2^{-1}$. The equality of the closed-interval cover infimum with Lebesgue outer measure in Countable covers by closed boxes, by open boxes and by closed cubes all compute Lebesgue outer measure therefore gives ${\lambda }_1^{\ast }(S)\ge 2^{-1}$, so $S$ is nowhere dense and not ${\lambda }_1$-null. The exact value ${\lambda }_1(S)=1\mathrm{/}2$ is computed on the companion page.

Why the transfer needs saying at all. The published items were written before any outer measure existed here, so they are stated as assertions about interval covers and cannot mention ${\lambda }_1$. Without the agreement theorem, a reader meeting both vocabularies would have two apparently unrelated notions of "measure zero" on the line; with it there is one notion, and the earlier refutations keep their force in the new vocabulary.