Conditions for convergence theorems with respect to monotone non-additive measure

Abstract

This paper discusses convergence theorems describing implications between six convergence concepts with respect to non-additive measure: almost everywhere convergence, pseudo-almost everywhere convergence, almost uniform convergence, pseudo-almost uniform convergence, convergence in measure, and convergence pseudo-in measure. The paper shows several new convergence theorems and arranges them together with existing convergence theorems by using ordinality and duality. In addition, it gives a new necessary condition and a new sufficient condition for the Egoroff theorem to hold.