Order Continuous Probabilistic Riesz Norms

Abstract

The concepts of order continuous norm, σ-order continuous norm, and Fatou norm defined on ordinary normed Riesz spaces are very important in the study of Riesz spaces. In this paper, we introduce the probabilistic analogues of such norms on a topological probabilistic normed Riesz (TPNR) space, and investigate their basic properties. In this context, some well-known theorems of the classical theory of topological Riesz spaces are proved in the setting of TPNR spaces, but now using the tools of probabilistic normed (PN) spaces. However, an interesting and different point here is that, although the classical order continuous Riesz norms are order preserving, the probabilistic Riesz norms considered in this work are order reversing mappings due to the nature of probabilistic distances.