# Equivalent definitions of $BV$ space and of total variation on metric measure spaces

## Ambrosio Luigi - Di Marino Simone

published
year: 2014
journal: J. Funct. Anal.
abstract: In this paper we introduce a definition of $BV$ based on measure upper gradients and prove the equivalence of this definition, and the coincidence of the corresponding notions of total variation, with the definitions based on relaxation of L^1 norm of the slope of Lipschitz functions or upper gradients. As in the previous work by the first author with Gigli and Savaré in the Sobolev case, the proof requires neither local compactness nor doubling and Poincaré

The paper is available on the cvgmt preprint server.