Anzellotti's pairing theory and the Gauss--Green theorem

In this paper we obtain a very general Gauss-Green formula for weakly differentiable functions and sets of finite perimeter. This result is obtained by revisiting Anzellotti's pairing theory and by characterizing the measure pairing $(\boldsymbol{A}, Du)$ when $\boldsymbol{A}$ is a bounded divergence measure vector field and $u$ is a bounded function of bounded variation.