Characterizing measurability, distribution and weak convergence of random variables in a Banach space by total subsets of linear functionals

AbstractConsider a generalized random variable X assuming values in a Banach space X with conjugate space X∗. For separable or reflexive X the measurability, probability distribution, and other properties of X are characterized in terms of a collection of real random variables {a∗(X) : a∗ ∈ A} and their linear combinations, where A is a total subset of X∗, i.e., A distinguishes points of X. Convergence in distribution of a sequence {Xn} is characterized in terms of uniform convergence of finite-dimensional distributions formed from {x∗(Xn) : x∗ ∈ X∗} (for X separable) or from {a∗(Xn) : a∗ ∈ A} (for X separable and reflexive). These results extend earlier ones known for the special cases A = X∗ or X = C[0, 1]. The proofs are based on theorems of Banach, Krein, and Šmulian characterizing the weak∗ closure of a convex set in X∗. An application to random Fourier series is given