The problem is whether every compact group has a subgroup which is not Haar measurable. This was known to be true for abelian groups [E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory. Berlin: Springer (1963; Zbl 0115.10603)]. The problem has subsequently been reduced to remain only for measurable infinite profinite groups (here denoted \(G\), profinite being projective limits of finite).
Solecki and Sternberg suggested in the last sentence of their article that the Hewitt and Ross result could be extended using cardinality methods. The present authors try to use the Continuum Hypothesis to prove the existence of a non-measurable infinite sub-group; if it has cardinality less than c, the cardinality of the continuum, the subgroup cannot be measurable.
S. Hernández et al. proved already in [J. Group Theory 19, No. 1, 179–189 (2016; Zbl 1332.22005)] that all measurable infinite profinite groups have non-measurable subgroups. The authors incorrectly assumed that all measurable profinite infinite groups are isomorphic as measure spaces to a particular Cantor group. This is an the infinite product of groups \(\mathbf{Z}_{2}\) and it is provided with Lebesgue measure. It corresponds to the middle thirds Cantor set and the use of symbolic dynamics with two symbols. Cantor groups can correspond to symbolic dynamics with more than two symbols. The reviewer conjectures that every measurable compact infinite profinite group has a subgroup which is a Cantor group. The authors have used a technique in their argument using what they call Markov sets. The reviewer asked the authors to provide a reference for their use of the term. One author refused to answer and the others did not respond. The reviewer has subsequently traced that A. Markoff, in a remarkable article in [Mat. Sb., Nov. Ser. 18(60), 3–28 (1946; Zbl 0061.04208)], discussing the analogy between sets in discrete groups and closed sets in topological groups, answering a problem he had encountered in 1945. He called these unconditionally closed sets as they are closed in any topology on the group.
The clever method of defining subgroups using cosets of a topological group had been used by B. E. Clark et al. [Topol. Proc. 10, No. 2, 251–257 (1985; Zbl 0625.22002)]. It usually involves products of \(G\) so include what the present authors call Fubini-Markov sets. All Markov and Fubini-Markov sets are shown to be null sets. The authors also provide a Baire category version of their theorem by substituting meagre sets instead of Markov sets.