The Gauss-Ostrogradskii formula in infinite-dimensional space

The aim of the present paper is to generalize the Gauss–Ostrogradskii theorem to an infinite-dimensional space $X$. On this space we consider not only Gaussian measures but a wider class of measures, differentiable along some Hilbert space continuously embedded in $X$. In the paper, a construction of a surface measure which employs ideas of the Malliavin calculus and the theory of Sobolev capacities is considered. It is a generalisation of the surface integration developed by Malliavin for the Wiener measure.