Fix an open set $\Omega \subset \mathbb{R}^n$. If $E$ is a measurable subset of $\Omega$, we may define the perimeter of $E$ in $\Omega$, denoted by $P(E;\Omega)$, to be
$$P(E;\Omega) = \sup_{\underset{||\phi||_{\infty} \leq 1}{\phi \in C^1_c(\Omega;\mathbb{R}^n)}} \int_E \mbox{div} \phi \, dx,$$
and when $P(E;\Omega) < +\infty$ we say that $E$ has finite perimeter in $\Omega$.
A basic property of any set of finite perimeter is that it may be approximated by sets with smooth boundary, in the following sense: for $E$ with $P(E;\Omega) < + \infty$, there exists a sequence $E_n$ such that $$\chi_{E_n} \xrightarrow{L^1} \chi_E$$ $$P(E_n; \Omega) \to P(E;\Omega),$$ where the $E_n = G_n \cap \Omega$ and the $G_n$ have smooth boundaries.
This is shown in Theorem 3.42 and Remark 3.43 of "Functions of Bounded Variation and Free Discontinuity Problems" by Ambrosio, Fusco, and Pallara. The basic argument involves first considering $\Omega = \mathbb{R}^n$, approximating $\chi_E$ by a sequence $\{u_n\}$ of smooth functions, and then defining $E_n = \{ u_n > t\}$ for an appropriate choice of $t$. The coarea formula, together with the lower-semicontinuity of the perimeter functional, can combine to prove that $P(E_n;\Omega) \to P(E;\Omega)$. The precise choice of $t$ is mostly irrelevant -- Sard's theorem implies that for almost every $0 < t < 1$ the resulting $E_n$ have smooth boundary. For more general $\Omega$, one needs some sort of extension operator (so there are some modest regularity conditions on $\Omega$ for this to be true).
My question is this: is it possible to do the same sort of approximation, but with the additional constraint $$m(E_n) = m(E),$$ where $m$ denotes Lebesgue measure? More specifically, are you aware of a result of this sort in the literature?