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Cylinder measures over vector spaces Millington, Hugh Gladstone Roy

Abstract

In this paper we present a theory of cylinder measures from the viewpoint of inverse systems of measure spaces. Specifically, we consider the problem of finding limits for the inverse system of measure spaces determined by a cylinder measure μ over a vector space X. For any subspace Ω of the algebraic dual X* such that (X,Ω) is a dual pair, we establish conditions on μ which ensure the existence of a limit measure on Ω . For any regular topology G on Ω, finer than the topology of pointwise convergence, we give a necessary and sufficient condition on μ for it to have a limit measure on Ω Radon with respect to G We introduce the concept of a weighted system in a locally convex space. When X is a Hausdorff, locally convex space, and Ω is the topological dual of X , we use this concept in deriving further conditions under which μ will have a limit measure on Ω Radon with respect to G. We apply our theory to the study of cylinder measures over Hilbertian spaces and ℓ(ρ)-spaces, obtaining significant extensions and clarifications of many previously known results.

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