On the existence of nonmeasurable subgroups of commutative groups. (Q1852444)

It is shown that, given a commutative group \(G\) of cardinality \(\omega_1\), there exists a countable family \({\mathcal G}\) of subgroups of \(G\) such that for each diffuse probability measure \(\mu\) on \(G\) some element of \({\mathcal G}\) is not \(\mu\)-measurable (Theorem 2). The proof uses an Ulam matrix and a representation of \(G\) as the union of an increasing sequence of some special subgroups of \(G\). An analogous result is also formulated for vector spaces over \(\mathbb{Q}\) of cardinality \(\omega_1\). The author asks if the following partial generalization of Theorem 2 holds: Given a commutative group \(G\) whose cardinality is not real-valued measurable and a diffuse probability measure \(\mu\) on \(G\), there exists a subgroup of \(G\) which is not \(\mu\)-measurable. Reviewer's remarks: (1) A similar question might be asked about other types of universal algebras, e.g., commutative semigroups. (2) Related results and methods are discussed in the author's book [``Selected topics of point set theory'', Wydawnictwo Uniwersytetu Łódzkiego, Łódź (1996), Section 5] and in his recent paper [Georgian Math. J. 10, No. 2, 247--255 (2003; Zbl 1044.28009)].