On weighted Hilbert spaces and integration of functions of infinitely many variables

We study aspects of the analytic foundations of integration and closely related problems for functions of infinitely many variables x(1), x(2), ... is an element of D. The setting is based on a reproducing kernel k for functions on D, a family of non-negative weights gamma(u), where u varies over all finite subsets of N, and a probability measure rho on D. We consider the weighted superposition K = Sigma(u) gamma(u)k(u) of finite tensor products k(u) of k. Under mild assumptions we show that K is a reproducing kernel on a properly chosen domain in the sequence space D-N, and that the reproducing kernel Hilbert space H (K) is the orthogonal sum of the spaces H (gamma(u)k(u)). Integration on H(K) cans be defined in two ways, via a canonical representer or with respect to the product measure rho(N) on D-N. We relate both approaches and provide sufficient conditions for the two approaches to coincide. (C) 2013 Elsevier Inc. All rights reserved.