Hilbert spaces of measures (Q1110891)

We continue the study of the structure of families of probability measures, started by \textit{I. Sh. Ibramkhalilov} and \textit{A. V. Skorokhod} [Estimates of parameters of stochastic processes (1980; Zbl 0429.60031)] and \textit{Z. S. Zerakidze} [Soobshch. Akad. Nauk Gruz. SSR 113, 37-39 (1984; Zbl 0562.60002)]. It turned out that in studying this question one can successfully apply the theory of Hilbert spaces. Here one must consider the linear spans of \(\sigma\)-additive measures of alternating sign, spanned by the original family of probability measures. One isolates the ``skeleton'' of such a family, establishes a representation of the Hilbert space of measures in the form of a direct sum of subspaces determined by a skeleton, studies linear functionals of integral type on Hilbert spaces of measures and gives a criterion for weak separability of measures of specific skeletons in terms of sets of such functionals.