Let \(\Gamma\) be a countably infinite abelian group and G its dual group. A subset S of \(\Gamma\) is called a w-set in \(\Gamma\) if there is a continuous complex-valued Borel measure \(\mu\) in G such that \(| {\hat \mu}| \geq 1\) everywhere in S. The author explains that w refers to Wiener (“Ka-set” was the term the reviewer has used for such sets in some of his papers thus referring to a previous work of Kaufman; apparently the author has not accepted this notation). The result of the paper under review is as follows: in the metric space \(2^{\Gamma}\) the class \(w\Gamma\) of all w-sets is an analytic set but not a Borel set. The analyticity is easy to prove, so that only “non-Borel” is essential in this theorem. The author reminds that the class of all Sidon sets in \(\Gamma\) is of type \(F_{\sigma}\); since every Sidon set is a w-set the theorem implies that not every w-set is Sidon. This is known and can be proved either by examples or by category method, as did the present author is one of his papers quoted in the references.
The proof is given separately for \(\Gamma\) such that for every \(m=1,2,3,..\). \(\Gamma\) /m\(\Gamma\) is finite (groups “of type I”) and for the opposite case (“groups of type II”). In both cases the proof is based on a theorem of Hurewicz saying that, for a (topological) Cantor set F, the uncountable closed subsets of F constitute an analytic but non-Borelian class in the space \(2^ F\) of all closed subsets of F with Hausdorff distance.
If \(\Gamma\) is of type I, G contains a Kronecker set K which may be assumed to be Cantor. The map B: 2\({}^ K\to 2^{\Gamma}\) is defined by \(B(E)=\{\chi \in \Gamma:| \chi -1| <1/3\) on \(E\}\). This map is lower semi-continuous and one to one (the author observes at the end of the paper that with some caution one can modify B so that it becomes a homeomorphism). Thus one has to prove that B(E) is a w-set if and only if E is uncountable. Then, \(B(2^ K)\) being a Borel set in \(2^{\Gamma}\), it remains but to apply Hurewicz’s theorem. Only the part “only if” is difficult. To prove this implication the author uses some results of Lindahl and Poulsen and of Rudin about the group of unimodular continuous functions on 0-dimensional compact metric space, especially the Bochner- type theorem for such groups.
For \(\Gamma\) of type II one must adapt the proof to the situation where there are no Kronecker sets in G but only \(K_ p\)-sets. So is called a set such that every continuous function on it to the group of p-th roots of unity is the restriction of a continuous character. In order to be able to apply Hurewicz’s theorem in this case one must define a suitable transformation analogous to B and this requires a technical device or even “a ruse” as the author calls it.