Journal of the Australian Mathematical Society - Series A
Vol. 64 Part 2 (1998)
T-regular probabilistic convergence spaces
A probabilistic convergence structure assigns a probability that a given filter converges to a given element of the space. The role of the t-norm (triangle norm) in the study of regularity of probabilistic convergence spaces is investigated. Given a probabilistic convergence space, there exists a finest T-regular space which is coarser than the given space, and is referred to as the `T-regular modification'. Moreover, for each probabilistic convergence space, there is a sequence of spaces, indexed by nonnegative ordinals, whose first term is the given space and whose last term is its T-regular modification. The T-regular modification is illustrated in the example involving `convergence with probability ' for several t-norms. Suitable function space structures in terms of a given t-norm are also considered.
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