An extension theorem for finite positive measures on surfaces of finite‎ ‎dimensional unit balls in Hilbert spaces

A consistency criteria is given for a certain class of finite positive measures on the surfaces of the finite dimensional unit balls in a real separable Hilbert space. It is proved, through a Kolmogorov type existence theorem, that the class induces a unique positive measure on the surface of the unit ball in the Hilbert space. As an application, this will naturally accomplish the work of Kanter (1973) on the existence and uniqueness of the spectral measures of finite dimensional stable random vectors to the infinite dimensional ones. The approach presented here is direct and different from the functional analysis approach in Kuelbs (1973), Linde (1983) and the indirect approach of Tortrat (1976) and Dettweiler (1976).