The Vitali-Hahn-Saks theorem and measuroids

Abstract

Abstract.We introduce the concept of measuroids, and prove the Uniform Boundedness Principle of Nikodym-Grothendieck, the Vitali-Hahn-Saks theorem and the Nikodym Convergence Theorem for measuroids.

Schur-type theorems for k-triangular lattice group-valued set functions with respect to filter convergence

“…for every ∈ . In correspondence with , there is ∈ ℱ * , ⊂ , satisfying (11). Note that the sequence ( )(ℕ), ∈ , does not (ℱ( ))converge to 0.…”

“…In [1,7] it is dealt with the -measures, that is monotone set functions with (∅) = 0, continuous from above and from below and compatible with respect to finite suprema and infima, which are a particular class of 1-triangular set functions and have several applications, for example to intuitionistic fuzzy events and observables (see also [1,10]). Observe that there are 1-triangular set functions which are not necessarily monotone, for instance the measuroids (see also [11]). In the context of -subadditive and/or -triangular lattice group-valued set functions, in [5] some Schur, Vitali-Hahn-Saks and Nikodým convergence theorems are proved with respect to the classical ( )-convergence, while in [4] some limit theorems are given with respect to filter convergence.…”

Equivalence between limit theorems for lattice group-valued k-triangular set functions

“…In particular we treat (s)-boundedness and continuity from above at ∅. Among the related literature, see for instance [17,24,25,26,28,30,31]. Some examples of k-triangular set functions are the so-called "M -measures", namely monotone set functions m with m(∅) = 0, continuous from above and from below and compatible with respect to finite suprema and infima, which have several applications, for example to intuitionistic fuzzy sets and observables (see also [2,27]).…”

Dieudonné-type theorems for lattice group-valued k-triangular set functions

“…Some examples of k-triangular set functions are the M -measures, that is monotone set functions m with m(∅) = 0, continuous from above and from below and compatible with respect to supremum and infimum, which have several applications in various branches, among which intuitionistic fuzzy sets and observables (see also [1,17,27,35,42]). Some examples of non-monotone 1-triangular set functions are the Saeki measuroids (see [43]). In [17,19,20,21,22,23] some limit theorems were proved for lattice group-valued k-subadditive capacities and k-triangular set functions.…”