Let $S\subset \mathbb{R}^n$ be a set of Hausdorff dimension $n-1$ and measure theoretic normal $\nu$. I will use $\mu$ to denote the restriction of $\mathcal{H}^{n-1}$ to $S$. That is, $\mu(E) = \mathcal{H}^{n-1}(E\cap S)$.
Suppose also that $f:S\to\mathbb{R}$ is a $\mu$-measurable function such that $$ \int_S |f(x)|\,d\mathcal{H}^{n-1} < \infty. $$
Does there necessarily exist a function with bounded variation $u\in BV(\mathbb{R}^n)$ such that the jump part of the total variation of $u$ is precisely $f\mu$?
I believe that the answer is yes when both $S$ and $f$ are smooth. For example, one can construct functions whose jump set is precisely the intersection of the hyperplane $\{x_n=0\}$ with the unit ball, and whose jump is constant. With a covering argument, this gives the result when $S$ and $f$ are smooth.
However, when $S$ is not smooth it seems much harder to patch together functions that have the correct jumps.