Generalized convergence in measure theorems of Sugeno integrals

Abstract

In this note, we show a generalized version of convergence in measure theorem of Sugeno integrals in the framework concerning ordered pair of monotone measures. The condition of a kind of absolute continuity we employed is not only sufficient, but also necessary for the generalized version. The previous two versions (standard-form and pseudo-form) of the convergence in measure theorem are recovered by this generalized version. Thus, the convergence in measure theorems of Sugeno integrals are unified in general framework. In the same way the convergence in measure theorem and strict convergence in measure theorem of seminnormed fuzzy integrals are generalized, respectively. The relations between the convergence in measure and the convergence pseudo-in measure of measurable functions sequence are described in the context relating to a pair of monotone measures.