A divergence theorem for Hilbert space

Abstract

Let B be a real separable Banach space. A suitable linear imbedding of a real separable Hilbert space into B with dense range determines a probability measure on B which is known as abstract Wiener measure. In this paper it is shown that certain submanifolds of B carry a surface measure uniquely defined in terms of abstract Wiener measure. In addition, an identity is obtained which relates surface integrals to abstract Wiener integrals of functions associated with vector fields on regions in B. The identity is equivalent to the classical divergence theorem if the Hilbert space is finite dimensional. This identity is used to estimate the total measure of certain surfaces, and it is established that in any space B there exist regions whose boundaries have finite surface measure.