On Borel measurability and large deviations for fuzzy random variables

Fuzzy random variables are a special case of measurable mappings taking on values in a non-separable metric space. Due to that non-separability, the two main definitions in the literature turn out not to be equivalent. We give an easy necessary and sufficient condition for their equivalence. An intermediate result of independent value is a characterization of separable subsets of that metric space. Several probabilistic applications are proposed, notably a generalization to this setting of the large deviations principle of Bolthausen.