On almost uniform convergence theorems for the smallest semicopula-based universal integral

Abstract

In this paper, we introduce a new property of a semicopula, called the uniform left (or right)-continuity in the first (or second) variable. Based on this new concept of continuity, a uniform convergence theorem for the smallest semicopula-based universal integral is given. In particular, a counter-example is presented to show that Theorem 2.9 in Borzová-Molnárová et al. (2015) [4] is not true. Finally, some modified versions of Theorems 2.7, 2.8 and 2.9 in Borzová-Molnárová et al. (2015) [4] are studied.