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The Central Limit Theorem problem for t-normed sums of random upper semicontinuous functions

We will present the problem of the Central Limit Theorem for random upper semicontinuous functions. Let us begin by providing the context for that problem. We need to present the space in which we will work, and specify the notions of random element of that space, convergence and the operations used to average random elements. Then we will make some historical remarks on limit theorems in this setting. 1. Space of u.s.c. functions. Amongst upper semicontinuous (u.s.c.) functions we refer, more precisely, to the space Fc of all functions u from a separable Banach space E to [0, 1] such that (a) Each level set uα = {x | u(x) ≥ α} is closed, convex and non-empty, for α ∈ (0, 1], (b) The support of u is relatively compact (we will denote its closure by u0). 2. Random u.s.c. functions. A random u.s.c. function is a mapping X from a measurable space (Ω,A) to Fc such that for each level mapping Xα, {ω ∈ Ω | Xα(ω) ∩G 6= Ø} ∈ A for every open set G ⊂ E. See [10, 3]. 3. Convergence. Convergence in Fc is meant in one of the following senses: