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In this paper, for a frameL, we characterize modified sobriety in stratifiedL-topological spaces and strongL-topological spaces internally using certain classes of stratifiedL-filters. While the first characterization using completely primeL-filters is trivial and applies to all stratifiedL-topological spaces, the other one using irreducibleL-filters generalizes an approach of R.E. Hoffmann to the lattice-valued case but is restricted to either the case that the lattice is a complete Boolean algebra or to the case of completely distributive lattices and strongL-topological spaces. [ABSTRACT FROM AUTHOR]

A random function of a variabletinT(or, a random function onT) is known as a functionfwhose values are random variables all defined on a common probability space, whereTis an arbitrary set. A random function is also called a stochastic (random) process. In this work, we base ourselves on a random function ofE-process type and such a function is also called a random function, briefly. In this approach, the domain of such a random function isor an interval of, and the set of values of this random function is considered as a special probabilistic metric (PM) space (more precisely, anE-space) of metric space-valued random variables, and all our definitions and results are presented using the tools of PM spaces. In this context, we introduce the concept of an exhaustive family of such random functions, which is a natural generalization of equicontinuity, and we investigate its basic properties. We also examine some of the properties related to the continuous convergence in probability for a sequence of such random functions and certain conditions which give rise to the continuity in probability of the limit of a sequence of such random functions. [ABSTRACT FROM PUBLISHER]

We develop a theory of probabilistic convergence spaces based on Tardiff’s neighbourhood systems for probabilistic metric spaces. We show that the resulting category is a topological universe and we characterize a subcategory that is isomorphic to the category of probabilistic metric spaces. [ABSTRACT FROM PUBLISHER]