Sets of finite perimeter and the Hausdorff-Gauss measure on the Wiener space. (English) Zbl 1196.46029
The integration by parts formula is studied in the abstract Wiener space. The author generalizes this formula from the case of finite-dimensional Euclidean space to sets of finite perimeter in infinite-dimensional spaces. For this, a concept of a measure theoretic boundary of a subset in an infinite-dimensional space is defined and studied. Technically, an increasing sequence of finite-dimensional subspaces with the union everywhere dense in a separable Banach space is utilized. Using intersections of a given subset with such linear subspaces and Hausdorff-Gauss measures, a measure theoretic boundary is defined. It may be strictly smaller than the topological boundary. Then, Sobolev spaces are applied. On the other hand, by its structure, this theory can also be considered as a generalization of Stokes’ formula. The integration by parts formula is written with respect to the one-codimensional Hausdorff-Gauss measure on the measure-theoretic boundary.
Reviewer: Sergey Ludkovsky (Moskva)
MSC:
| 46G12 | Measures and integration on abstract linear spaces |
| 28A75 | Length, area, volume, other geometric measure theory |
| 26E15 | Calculus of functions on infinite-dimensional spaces |
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