On vector-valued Dobrakov submeasures

Abstract

Ivan Dobrakov has initiated a theory of non-additive set functions defined on a ring of sets intended to be a non-additive generalization of the theory of finite non-negative countably additive measures. These set functions are now known as the Dobrakov submeasures. In this paper we extend Dobrakov's considerations to vector-valued submeasures defined on a ring of sets. The extension of such submeasures in the sense of Drewnowski is also given. IntroductionNon-additive set functions, as for example outer measures, semi-variations of vector measures, appeared naturally earlier in the classical measure theory concerning countable additive set functions or more general finite additive set functions. A systematic study of non-additive set function begins in the fifties of the last century, cf. [5]. Thence many authors have investigated different kinds of non-additive set functions, as submeasures [9], t-norms and t-conorms [18], k-triangular set functions [2] and null-additive set functions [25], fuzzy measures and integrals [12,24] and many other types of set functions and their properties. Specially, in different branches of mathematics as potential theory, harmonic analysis, fractal geometry, functional analysis, theory of nonlinear differential equations, theory of difference equations and optimizations, etc., there are many types of non-additive set functions.An interesting non-additive set function (as a generalization of a notion of submeasure) was introduced by I. Dobrakov. Definition 1.1 (Dobrakov, [6]) Let R be a ring of subsets of a set T = ∅. We say that a set function µ : R → [0, +∞) is a submeasure, if it is (1) monotone: if A, B ∈ R, such that A ⊂ B, then µ(A) ≤ µ(B);(2) continuous at ∅ (shortly continuous): for any sequence (A n ) ∞ 1 of sets from R, such that A n ց ∅ (i.e., A n ⊃ A n+1 for each n ∈ N and n∈N A n = ∅) there holds µ(A n ) → 0 as n → ∞;(3) subadditively continuous: for every A ∈ R and ε > 0 there exists a δ > 0, such that for every B ∈ R with µ(B) < δ there holds (a) µ(A ∪ B) ≤ µ(A) + ε, and (b) µ(A) ≤ µ(A \ B) + ε.1 Mathematics Subject Classification (2010): 28B05, 28B15 Key words and phrases: Non-additive set function, Dobrakov submeasure, L-normed Banach lattice, vector-valued measure, extension of a measure. Acknowledgement. This paper was supported by Grants VEGA 2/0097/08 and VVGS 45/10-11.

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