The space of functions of bounded variation on curves in metric measure spaces

Abstract:

The approximation modulus, $AM$–modulus for short, was defined in an earlier paper by the author. In this paper it is shown that an $L^1(X)$–function $u$ in a metric measure space $(X,d,\nu )$ can be defined to be of bounded variation on $AM$–a.e. curve in $X$ without an approximation of $u$ by Lipschitz or Newtonian functions. The essential variation of $u$ on $AM$ almost every curve is bounded by a sequence of non-negative Borel functions in $L^1(X)$. The space of such functions is a Banach space and there is a Borel measure associated with $u$. For $X = \mathbf {R}^n$ this gives the classical space of BV functions.