Measuring sets with translation invariant Borel measures

TL;DR

This paper investigates the structure of measured sets in the real line and Polish groups, showing that certain Borel sets can be decomposed into measured sets, while others cannot be expressed as countable unions of measured sets.

Contribution

It demonstrates that every Borel nullset of the second category can be partitioned into measured sets and constructs examples of sets not expressible as countable unions of measured sets.

Findings

Every Borel nullset of the second category is a union of two measured sets.

Non-locally compact Polish groups are not unions of countably many measured sets.

There exist measured sets null or non--sigma-finite for all Hausdorff measures.

Abstract

Following Davies, Elekes and Keleti, we study measured sets, i.e. Borel sets $B$ in $\mathbb{R}$ (or in a Polish group) for which there is a translation invariant Borel measure assigning positive and \sigma-finite measure to $B$. We investigate which sets can be written as a (disjoint) union of measured sets. We show that every Borel nullset $B\subset \mathbb{R}$ of the second category is larger than any nullset $A\subset \mathbb{R}$ in the sense that there are partitions $B=B_1\cup B_2$, $A=A_1\cup A_2$ and gauge functions $g_1, g_2$ such that the Hausdorff measures satisfy $H^{g_i}(B_i)=1$ and $H^{g_i}(A_i)=0$ ($i=1,2$). This implies that every Borel set of the second category is a union of two measured sets. We also present Borel and compact sets in $\mathbb{R}$ which are not a union of countably many measured sets. This is done in two steps. First we show that non-locally compact Polish groups are not a union of countably many measured sets. Then, to certain Banach spaces we associate a Borel and/or \sigma-compact additive subgroup of $\mathbb{R}$ which is not a union of countably many measured sets. It is also shown that there are measured sets which are null or non-\sigma-finite for every Hausdorff measure of arbitrary gauge function.