AM‐modulus and Hausdorff measure of codimension one in metric measure spaces. Issue 1 (10th January 2022) - Honzlová‐Exnerová, Vendula; Malý, Jan; Martio, Olli
Abstract: Let Γ ( E ) Γ (E) be the family of all paths which meet a set E in the metric measure space X . The set function E ↦ A M ( Γ ( E ) ) E AM(Γ (E)) defines the A M AM ‐modulus measure in X where A M AM refers to the approximation modulus [22]. We compare A M ( Γ ( E ) ) AM(Γ (E)) to the Hausdorff measure c o H 1 ( E ) coH^1(E) of codimension one in X and show that c o H 1 ( E ) ≈ A M ( Γ ( E ) ) equation*6pc coH^1(E) ≈ AM(Γ (E))-6pc equation* for Suslin sets E in X . This leads to a new characterization of sets of finite perimeter in X in terms of the A M AM ‐modulus. We also study the level sets of B V BV functions and show that for a.e. t these sets have finite c o H 1 coH^1 ‐measure. Most of the results are new also in R n R^n .