Decomposition of dyadic measures and unions of closed $\mathscr{U}$-sets for series in a Haar system

New properties of finitely additive set functions (quasi-measures) and Borel measures on dyadic product groups $\mathbb{G}^m$ are established. The results obtained are applied to the theory of series in Haar systems — for example, for a broad family of classes of multiple Haar series on $\mathbb{G}^m$, a countable union of closed uniqueness sets is shown to be a uniqueness set too.
Bibliography: 18 titles.