*Published Paper*

**Inserted:** 7 jun 2000

**Last Updated:** 20 feb 2008

**Journal:** J. Math. Pures Appl.

**Volume:** (9) 82

**Number:** 8

**Pages:** 975-1004

**Year:** 2003

**Abstract:**

In this paper we give a possible definition of the space of Banach space valued $BV$ functions on metric spaces; the metric space is supposed to be doubling and that it supports a PoincarĂ© inequality. The idea of the definition of $BV$ functions is to take the closure with respect to a suitable convergence of regular functions, the Lipschitz functions. The main problem with this definition is the proof that the total variation is a measure, and the techniques used are typical of the relaxation analysis.

In this paper we also define the sets of finite perimeter and we give some basic properties of this family of sets; the main tool that we prove in this section is the Coarea formula for $BV$ functions.

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