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 infinite many variables x1, x2, . . . ∈ D. The setting is based on a reproducing kernel k for functions on D, a family of non-negative weights γu, where u varies over all finite subsets of N, and a probability measure ρ on D. We consider the weighted superposition K = ∑ u γuku of finite tensor products ku of k. Under mild assumptions we show that K is a reproducing kernel on a properly chosen domain in the sequence space D, and that the reproducing kernel Hilbert space H(K) is the orthogonal sum of the spaces H(γuku). Integration on H(K) can be defined in two ways, via a canonical representer or with respect to the product measure ρ on D. We relate both approaches and provide sufficient conditions for the two approaches to coincide. Dedicated to J. F. Traub and G. W. Wasilkowski on the occasion of their 80th and 60th birthdays