Abstract
Let (Ω, A, P) be a probability space and E be a Banach space. We study the approximation of an E-valued random variable X, which is an element of the Orlicz space LΦ(Ω, A, P; E), by a function Y∃LΦ, which is measurable with respect to a sub-σ-field of A and takes values in a closed convex subset of E. Two types of approximation are considered: ∫Φ(∥X − Y∥) dP=inf, and NΦ(X−Y)=inf with the Orlicz space norm NΦ. We give conditions for the existence of best approximants. If E is reflexive, we obtain martingale type convergence theorems for best approximants and discuss the continuity of the operator X → best approximant of X.
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