# Countable vs. ultra-negligible sets [duplicate]

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 ?

• You're right. Sorry for that! Apr 7, 2014 at 16:45

• 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). Apr 8, 2014 at 1:13