Functions of bounded variation and the [formula omitted]-modulus in [formula omitted

Abstract

Moduli of path families are widely used to study Sobolev functions. Similarly, the recently introduced approximation (AM-) modulus is helpful in the theory of functions of bounded variation (BV) in Rn (Martio, 2016). We continue this direction of research. Let ΓE be the family of all paths which meet E⊂Rn. We introduce the outer measure E↦AM(ΓE) and compare it with other (n−1)-dimensional measures. In particular, we show that AM(ΓE)=2Hn−1(E) whenever E lies on a countably (n−1)-rectifiable set. Further, we study functions which have bounded variation on AM-a.e. path and we relate these functions to the classical BV functions which have only bounded essential variation on AM-a.e. path. We also characterize sets E of finite perimeter in terms of the AM-modulus of two path families naturally associated with E.