  A new notion of pairing between measure vector fields with divergence measure
and scalar functions, which are not required to be weakly differentiable, is
introduced. In particular, in the case of essentially bounded
divergence-measure fields, the functions may not be of bounded variation. This
naturally leads to the definition of $BV$-like function classes on which these
pairings are well defined. Despite the lack of fine properties for such
functions, our pairings surprisingly preserve many features of the recently
introduced $\lambda$-pairings (Crasta, De Cicco, Malusa 2022,
arXiv:1902.06052), as coarea formula, lower semicontinuity, Leibniz rules, and
Gauss-Green formulas. Moreover, in a natural way new anisotropic "degenerate"
perimeters are defined, possibly allowing for sets with fractal boundary.
