How is the surface measure defined when it comes to integrals over the boundary of some open, set?

Question

I’m reading Evans’ PDEs book and on appendix C (Calculus), he defines an integral with the so-called “surface measure”. I think by that it’s implied that he’s defining the integral in the measure-theoretic way. However, I’m not sure how this measure is defined to begin with. The book doesn’t specify it either. Could anyone clarify for me how is the surface measure defined and, in particular, how to make sense of the corresponding integrals? What I do know is that the functions over which we are integrating are defined on some open set in $\mathbb{R}^n$, have compact support, and are usually at least $C^2$.