Measures on linear topological spaces

This survey contains some results on measures in linear topological spaces and in completely regular topcdogical spaces. These results are important in the theory of linear differential equations involving functions of an infinite-dimensional argument. We give conditions for the countable additivity of signed (and, more generally, vector-valued) cylindrical measures on products of measurable spaces (“Kolmogorov's theorem”) and for (again, signed and even vector–valued) cylindrical measures on arbitrary separated locally convex spaces (“Minlos–Sazonov theorem”, proved for signed measures by Shavgulidze). We consider the connection between Radon measures defined on $\sigma$-algebras of Borel subsets of a completely regular topological space and certain linear functionals on the space of bounded continuous real-valued functions defined on such a space. We describe a number of classes of completely regular topological spaces $X$ having the property that the standard Prokhorov condition turns out to be sufficient or necessary for the relative weak sequential compactness of sets of Radon measures on $X$. For the case of positive Radon measures defined on locally convex spaces we prove a “P. Lévy theorem” and several related assertions, in which conditions for the weak convergence of sequences of positive measures and conditions for the weak compactness of families of such measures are stated as conditions on the families of their Fourier transforms.