T-REGULAR PROBABILISTIC CONVERGENCE SPACES

A probabilistic convergence structure assigns a probability that a giv en filter converges to a given element of the space. The role of the t -norm (triangle norm) in the study of regularity of probabilistic conv ergence spaces is investigated. Given a probabilistic convergence spac e, there exists a finest T-regular space which is coarser than the giv en space, and is referred to as the 'T-regular modification'. Moreover , for each probabilistic convergence space, there is a sequence of spa ces, indexed by nonnegative ordinals, whose first term is the given sp ace and whose last term is its T-regular modification. The T-regular m odification is illustrated in the example involving 'convergence with probability lambda' for several t-norms. Suitable function space struc tures in terms of a given t-norm are also considered.