On the compactness criterion for probability measures on Banach spaces (Q1072209)

Let X be a real separable Banach space and \(X^*\) its topological dual space. Let \({\mathcal P}(X)\) denote the set of all probability measures on X with the Prokhorov metric. For each \(\mu\in {\mathcal P}(X)\), let \({\hat \mu}\) denote the characteristic functional of \(\mu\). \textit{Yu. V. Prokhorov} [Teor. Veroyatn. Primen. 1, 177-237 (1956; Zbl 0075.290); English translation in Theory Probab. Appl. 1, 157-214 (1956)] gave a compactness criterion for a set of probability measures on a real separable Hilbert space in terms of their characteristic functionals. In this paper, the author considers the following assertion which corresponds to Prokhorov's result when X is a Hilbert space: A subset K of \({\mathcal P}(X)\) is relatively compact if (and only if) for each \(\epsilon >0\), there exists a family \(\{S_{\mu,\epsilon}\); \(\mu\in K\}\) of positive symmetric nuclear operators from \(X^*\) into X which satisfies the following two conditions: (i) 1-Re \({\hat \mu}\)(f)\(\leq <S_{\mu,\epsilon}f,f>+\epsilon\) for all \(f\in X^*\) and all \(\mu\in K,\) (ii) the set \(\{S_{\mu,\epsilon}\); \(\mu\in K\}\) is relatively compact with respect to the nuclear form. Using a result of \textit{S. Kwapień} [Stud. Math. 44, 583-595 (1972; Zbl 0256.46024)], the author proves that the above assertion is not valid unless X is isomorphic to a Hilbert space.