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 ‎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 the work of Kuelbs and Linde and‎ ‎the indirect approach of Tortrat and Dettweiler‎.