Extension of Gaussian cylinder set measures

Abstract

Let E be a one-to-one continuous map of the real and separable Hilbert space H into the real and separable Hilbert space K , with E having dense range. One considers Gaussian cylinder set measures on H defined by weak covariance operators. Such cylinder set measures may be used to induce, through E , Gaussian cylinder set measures on K . The result of this paper extends a result of Sato: it characterizes the norm of the spaces K for which the induced measure extends to a probability measure on the Borel sets of K . Such a result is of interest in the robustness study of signal detection.