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A generalization of a theorem of Dieudonné for k-triangular set functions. (English) Zbl 0609.28002
As it is well-known, the Nikodým boundedness theorem for measures in general fails for algebras of sets. But there are uniform boundedness theorems in which the initial boundedness conditions are on some subfamilies of a given \(\sigma\)-algebra; those subfamilies may not be \(\sigma\)-algebras. A famous theorem of Dieudonné states that for compact metric spaces the setwise boundedness of a family of Borel regular measures on open sets implies its uniform boundedness on all Borel sets. In this paper Dieudonnés theorem on a wider class of set functions (in general non-additive) is generalized. The main result is Theorem 2.: Let \({\mathcal M}\) be a family of k-triangular set functions defined on the collection \({\mathcal B}\) of all Borel sets of a Hausdorff locally compact topological space T with regular variations. If the set \(\{\mu (0);\quad \mu \in {\mathcal M}\}\) is bounded for every open set O, then \(\{\mu (B);\mu \in {\mathcal M},\quad B\in {\mathcal B}\}\) is a bounded set. With some modifications also a generalization of Dieudonné’s theorem for semigroup valued set functions is obtained.

28A10 Real- or complex-valued set functions
28B10 Group- or semigroup-valued set functions, measures and integrals
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)