Borel sets which are null or non-$\sigma$-finite for every translation invariant measure

TL;DR

This paper demonstrates that certain Borel sets, including Liouville numbers and others, are either null or non-$\sigma$-finite for all translation-invariant measures, answering a longstanding question and exploring their measure-theoretic properties.

Contribution

It establishes that specific Borel sets are either null or non-$\sigma$-finite for all translation-invariant measures, including Hausdorff measures, and characterizes their measure-theoretic complexity.

Findings

Liouville numbers are null or non-$\sigma$-finite for all translation-invariant measures.

Certain Borel sets like non-normal numbers also share this property.

The measure-theoretic complexity of these sets can vary widely, with arbitrary dimensions.

Abstract

We show that the set of Liouville numbers is either null or non-$\sigma$-finite with respect to every translation invariant Borel measure on $\RR$, in particular, with respect to every Hausdorff measure $\iH^g$ with gauge function $g$. This answers a question of D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch-Eggleston numbers, as well as all Borel subgroups of $\RR$ that are not $F_\sigma$ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.