The smallest semicopula-based universal integrals II: Convergence theorems
Section snippets
Introduction and description of results
This is the second part of the two-paper set devoted to a detailed study of the weakest semicopula-based universal integrals. Both are motivated by the recent ideas of universal integrals in the sense of [3] which can be defined for arbitrary measurable spaces, arbitrary monotone measures and arbitrary measurable functions.
In particular, we consider measurable spaces , where is a σ-algebra of subsets of X, and we denote by the class of all measurable spaces. For a given measurable
Monotone convergence
Throughout this text we use the notation ↗ for a non-decreasing sequence and the notation ↘ for a non-increasing sequence (of functions, or sets). Under the notation we understand a pointwise convergence of a sequence of (measurable) functions to a (measurable) function f on an appropriate set, in our case on usually.
The following elementary proof of implication follows the one provided in [1] for the Sugeno integral case. Theorem 2.1 Monotone convergence I Let be left-continuous and .
Concluding remarks
We have investigated convergence theorems of semicopula-based universal integrals . We have shown necessary and sufficient conditions for convergence of integral sequences with respect to different types of convergences of measurable functions: almost everywhere convergence (m-a.e.), almost uniform (m-a.u.), convergence in measure (m), strict convergence in measure (s-m), and convergence in mean (). It emerges that each type of convergence is related to a certain structural characteristic
Acknowledgements
Authors thank Radko Mesiar for providing useful comments and improvements of previous versions of manuscript. In fact, the observation on C-universal integrals presented in Concluding remarks is due to him. The anonymous referees made many useful suggestions which were incorporated into the paper with gratitude. The support of the grants VVGS-PF-2014-453, VVGS-2013-121, and VEGA 1/0171/12 is kindly announced.
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