Convergence of a sequence of weakly regular set functions

Abstract

ABSTRACT. The present paper is devoted to generalizations of the Dieudonnd theorem claiming that the convergence of sequences of regular Borelian measures is preserved under the passage from a system of open subsets of a compact metric space to the class of all Borelian subsets of this space. The Dieudonnd theorem is proved in the case for which the set functions are weakly regular, nonadditive, defined on an algebra of sets that contains the class of open subsets of an arbitrary a-topological space, and take values in a uniform space.

On Drewnowski lemma for non-additive functions and its consequences

“…whenever τ ≡ τ U , following [17], we say that (iii) ϕ is s-outer if for every U ∈ U there exists V = V (U ) ∈ U such that for every orthogonal pair (a, b) of elements of L it holds…”

“…In the present paper, starting from the study of non-additive functions which take their values into some uniform as well as topological space and satisfy some additional properties ( [9,11,22] and more recently [17,18,26,27]), we establish an extended version of Drewnowski lemma for s-outer functions defined on an orthomodular lattice with the Subsequential Interpolation Property (see Lemma 4.1) which improves the available results in the literature, in particular those contained in [3,13] as shown in Remark 4.2. Moreover, jointly with Theorem 5.2, it enables us to deduce Brooks-Jewett and Cafiero theorems for such non-additive functions in Sect.…”

On Brooks–Jewett, Vitali–Hahn–Saks and Nikodým convergence theorems for quasi-triangular functions

“…As a consequence, the additivity assumption has to be replaced by a considerably weaker assumption, and various additional conditions have to be required in order to introduce special kinds of non-additive measures. The present paper is concerned with the class of quasi-triangular functions-see, e.g., [5,6,7,19,20,26,27,28]. The importance of this class of functions mainly stems from the following two facts.…”