Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Measuring sets with translation invariant Borel measures   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.

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The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph