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

Abstract

Let Γ(𝐸) be the family of all paths which meet a set 𝐸 in the metric measure space 𝑋. The set function 𝐸 ↦ 𝐴𝑀(Γ(𝐸)) defines the 𝐴𝑀-modulus measure in 𝑋 where 𝐴𝑀 refers to the approximation modulus [22]. We compare 𝐴𝑀(Γ(𝐸))to the Hausdorff measure 𝑐𝑜 1 (𝐸) of codimension one in 𝑋 and show that 𝑐𝑜 1 (𝐸) ≈ 𝐴𝑀(Γ(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 𝑐𝑜 1 -measure. Most of the results are new also in ℝ 𝑛 .