Algebraic, metric and probabilistic properties of convex combinations based on the t-normed extension principle: the strong law of large numbers

Consider the Strong Law of Large Numbers for t-normed averages of fuzzy random variables in the uniform metric d∞. That probabilistic property is known to hold when the t-norm is the minimum and to fail when the t-norm is the product. We prove that it is characterized by an algebraic property of the t-norm (that of being eventually idempotent) and by a metric property of the space of fuzzy sets (that it becomes a convex combination space). We show that the equivalence holds not only for Euclidean or Banach spaces, but in the more general setting of convex combination spaces.