Measures on General Codimensional Surfaces in Infinite Dimensions and Stokes-Type Theorems

Abstract

In this paper, we give an explicit construction of surface measures on a class of surfaces with arbitrary, possibly infinite, codimension in $\ell^2$. These measures are constructed from local representations associated with a fixed Gaussian product measure. We then establish local and global Gauss--Green-type formulas, introduce a notion of top-degree differential form, and derive an associated Stokes-type identity. We also determine the orientation of the boundary induced by the orientation of the surface. Moreover, therelationship between $\mathcal F$-continuity and Borel measurability is examined,which reveals a phenomenon specific to the infinite-dimensional setting.