Consider some open subset $U$ of $\mathbb{R}^n$ where $U$ has a (piecewise) $C^1$-boundary. Let $f$ be some smooth (enough) real function. Is there some way to give a measure-theoretic definition of the surface integral $\int_{\partial U} f dS$ without referring to methods from differential geometry, e.g. without mentioning $n$-forms and things like exterior derivatives. The only obvious condition is that a possible measure-theoretic definition, if it exists, coincides with the usual differential geometric definition for subsets of $\mathbb{R}^n$. I only need such a definition for reasonable subsets of $\mathbb{R}^n$. So, at least for me, it doesn't concern to (subsets of) general manifolds.
Kindly apreciated,
Aris