The Wiener Sphere and Wiener Measure

Abstract

The Loeb measure construction of nonstandard analysis is used to define uniform probability $\mu_L$ on the infinite-dimensional sphere of Poincare, Wiener and Levy, and we construct Wiener measure from it, thus giving rigorous sense to the informal discussion by McKean. From this follows an elementary proof of a weak convergence result. The relation to the infinite product of Gaussian measures is studied. We investigate transformations of the sphere induced by shifts and the associated transformations of $\mu_L$. The Cameron-Martin density is derived as a Jacobian. We also prove a dichotomy theorem for the family of shifted measures.