Decomposition of dyadic measures and unions of closed \( \mathcal{U}\)-sets for series in a Haar system (Q2817573)

In the paper under review, the authors prove new properties of finitely additive set functions (quasi-measures) and Borel measures on dyadic product groups \(G^m\). The main contributions in the paper are as follows: In Theorem 3.1, it is proved that any nontrivial quasi-measure majorizes some nontrivial nonnegative quasi-measure. Some interesting results on the decomposition of nonnegative quasi-measures on the group \(G^m\) which are concentrated on a union of closed sets can be found in Theorem 4.2 and Theorem 4.3. The authors define a Haar series on the group \(G^m\) and introduce various types of convergence of such series, and prove some auxiliary results. The authors also study Bari's theorem for one-dimensional and multiple Haar series. More precisely, they give a Bari-type theorem in Theorem 6.3 of the paper, on the union of uniqueness sets for series in classes with the hereditary property. Examples of such classes are also provided.