Elsevier

Fuzzy Sets and Systems

Volume 271, 15 July 2015, Pages 18-30
Fuzzy Sets and Systems

The smallest semicopula-based universal integrals II: Convergence theorems

https://doi.org/10.1016/j.fss.2014.09.024Get rights and content

Abstract

In this paper we continue studying the smallest universal integral IS having S as the underlying semicopula. We present convergence theorems for IS-integral sequences including monotone, almost everywhere, almost uniform, in measure and in mean converging sequences of measurable functions, respectively. It emerges that these convergences characterize the underlying measure properties such as null-additivity, monotone autocontinuity and autocontinuity. We provide many examples and counter-examples as well as a few interesting open problems.

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 (X,A), where A is a σ-algebra of subsets of X, and we denote by S 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 fnf we understand a pointwise convergence of a sequence of (measurable) functions (fn)1 to a (measurable) function f on an appropriate set, in our case on X=[0,1] usually.

The following elementary proof of implication (i)(ii) follows the one provided in [1] for the Sugeno integral case.

Theorem 2.1 Monotone convergence I

Let SS be left-continuous and mM(X,A)1.

Concluding remarks

We have investigated convergence theorems of semicopula-based universal integrals IS. 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 (IS). 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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