AM-modulus and Hausdorff measure of codimension one in metric measure spaces

Abstrakt

Let Gamma(E) be the family of all paths which meet a set E in the metric measure space X. The set function E bar right arrow AM (Gamma(E)) defines the AM-modulus measure in X where AM refers to the approximation modulus [22].

We compare AM (Gamma(E)) to the Hausdorff measure coH(1) (E) of codimension one in X and show that coH(1)(E) approximate to AM(Gamma(E)) for Suslin sets E in X. This leads to a new characterization of sets of finite perimeter in X in terms of the AM-modulus.

We also study the level sets of BV functions and show that for a.e. t. these sets have finite coH(1)-measure. Most of the results are new also in R-n.

Klíčová slova

• AM-modulus
• level sets of BV-functions
• metric measure spaces
• perimeter
• sets of co-dimension one