Gauss-Green formulas for divergence measure tensor fields on rough domains

We introduce a notion of pairing between essentially bounded tensor fields with divergence measure and vector-valued functions of bounded variation, extending the classical theory to the tensorial setting. This naturally leads to an adaptation of the definition of normal trace for tensor fields with measure divergence even on a rectifiable set. As a consequence, we establish tensorial Gauss-Green formulas that remain valid on sets with low regularity, including sets of finite perimeter. These results yield a unified and robust framework for integration by parts in the presence of irregular tensor fields and domains.