Approach properties of probabilistic metrizable approach spaces

Our investigation of approach properties in probabilistic metrizable approach spaces is based on two faithful functors. The first one was introduced in [9] and goes from probabilistic metric spaces with respect to a continuous t-norm to the category of approach spaces. The second one goes from probabilistic metric spaces with respect to a continuous t-norm to the category of uniform gauge spaces. Using these functors we show that all probabilistic metrizable approach spaces are uniform. We characterize those probabilistic metrizable approach spaces that are associated with a certainly bounded probabilistic metric space or by one that is bounded in distribution. We show that precompactness of the probabilistic metric space is equivalent to the associated uniform gauge space having zero index of precompactness, and that completeness of the probabilistic metric space is equivalent to completeness of the associated uniform gauge space.