A subset $A\subset\mathbb{R}$ is negligible if for each $\epsilon>0$ there exists a sequence $(I_n)$ of intervals such that $A\subset\cup_n I_n$ and $\sum_n \vert I_n \vert \leq \epsilon$. Let us say that $A$ is ultra-negligible if for any sequence $(\epsilon_n)$ of positive numbers, there exists a sequence $(I_n)$ of intervals such that $A\subset\cup_n I_n$ and $\vert I_n\vert \leq \epsilon_n$ for all $n$ (for example, any countable set is ultra-negligible). Clearly, every ultra-negligible set is negligible. In his famous problem book, P. Halmos shows that the converse is not true : the triadic Cantor set is negligible but not ultra-negligible.

I am wondering if there are examples of ultra-negligible sets that are not countable ?

  • $\begingroup$ You're right. Sorry for that! $\endgroup$ – MassiveJack Apr 7 '14 at 16:45

The sets you refer to as "ultranegligible" are known as the strong measure zero sets, and the assertion that every strong measure zero set is countable is known as the Borel conjecture, and is independent of ZFC.

If the continuum hypothesis holds (also if Martin's axiom holds), then there are uncountable strong measure zero sets, and so it is consistent with ZFC that the Borel conjecture fails. Meanwhile, Richard Laver proved the relative consistency of the Borel conjecture. And so it is independent of ZFC.

It happens that Kameryn Williams just gave a talk on this topic last Friday at the CUNY set theory seminar.

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  • $\begingroup$ To add more references, Arnold Miller has a paper Special subsets of the real line (e.g. see Theorem 13) at his web page (and some other papers about $\gamma$-sets might be related). $\endgroup$ – Mirko Apr 8 '14 at 1:13

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