How to prove measure theoretically that $\int_U u_{x_i} dx = \int_{\delta U} u \nu^i dS$ (Gauss-Green Theorem)?

I've been studying PDEs and one important technique is integration by parts, due to which, given functions $u, v \in C^1(\overline{U})$, we have$$\int_U u_{x_i} v dx = -\int_U uv_{x_i} dx + \int_{\delta U} uv \nu^i dS $$ for each $1 \leq i \leq n$. However, I came to the realization that I don't understand how integration by parts works in higher dimension. My understanding is that the integral on the left hand side is a Lebesgue integral in general, and the integrals on the right hand side are, respectively, a Lebesgue integral and another integral with respect to a different measure S. Is that right? If so, what measure is that? How can I prove from first principles (i.e., basic definitions of integrals in terms of sups) the above formula?