Equivalent definitions of $BV$ space and of total variation on metric measure spaces
J. Funct. Anal.
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
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