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 x1,x2,…∈Dx1,x2,…∈D. The setting is based on a reproducing kernel kk for functions on DD, a family of non-negative weights γuγu, where uu varies over all finite subsets of NN, and a probability measure ρρ on DD. We consider the weighted superposition K=∑uγukuK=∑uγuku of finite tensor products kuku of kk. Under mild assumptions we show that KK is a reproducing kernel on a properly chosen domain in the sequence space DNDN, and that the reproducing kernel Hilbert space H(K)H(K) is the orthogonal sum of the spaces H(γuku)H(γuku). Integration on H(K)H(K) can be defined in two ways, via a canonical representer or with respect to the product measure ρNρN on DNDN. We relate both approaches and provide sufficient conditions for the two approaches to coincide.