A new necessary and sufficient condition for convergence of generalized Sugeno integrals

Abstract

In this paper, we investigate convergence in measure for a sequence of generalized Sugeno integrals with respect to monotone measures. While the existing results typically require continuity of the underlying measure or provide only sufficient conditions, we establish new necessary and sufficient conditions that differ from the assumptions previously considered in the literature. We prove that the autocontinuity of a monotone measure completely characterizes convergence in measure for a sequence of generalized Sugeno integrals induced by a continuous semicopula. Our results are first established for the case where the range of the limiting function is finite, and are subsequently extended to classes of functions with infinite range under an additional assumption. In particular, the proof techniques developed in this paper are entirely different from those employed in the existing literature.