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

Details Borel sets which are null or non-$σ$-finite for every translation invariant measure Borel sets which are null or non-$σ$-finite for every translation invariant measure

2011-09-24

arxiv.org/abs/1109.5309v1

Adv. Math. 201 (2006), 102-115

Mathematics

Classical Analysis and ODEs

Scientific paper

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.

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