Sets of finite perimeter and the Hausdorff–Gauss measure on the Wiener space
نویسنده
چکیده
In Euclidean space, the integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a type of surface measure. According to geometric measure theory, this surface measure is realized by the one-codimensional Hausdorff measure restricted on the reduced boundary and/or the measure-theoretic boundary, which may be strictly smaller than the topological boundary. In this paper, we discuss the counterpart of this measure in the abstract Wiener space, which is a typical infinite-dimensional space. We introduce the concept of the measure-theoretic boundary in the Wiener space and provide the integration by parts formula for sets of finite perimeter. The formula is presented in terms of the integration with respect to the one-codimensional Hausdorff–Gauss measure restricted on the measure-theoretic boundary.
منابع مشابه
Sets of Finite Perimeter and Geometric Variational Problems An Introduction to Geometric Measure Theory
Contents Preface page xiii Notation xvii PART I RADON MEASURES ON R n 1 1 Outer measures 4 1.1 Examples of outer measures 4 1.2 Measurable sets and σ-additivity 7 1.3 Measure Theory and integration 9 2 Borel and Radon measures 14 2.1 Borel measures and Carathéodory's criterion 14 2.2 Borel regular measures 16 2.3 Approximation theorems for Borel measures 17 2.4 Radon measures. Restriction, supp...
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