Let $\nu$ be a finite Radon measure on $\mathbb{R}^2$ and denote the Lebesgue measure on $\mathbb{R}^2$ by $\mathcal{L}^2$. Assume that $\nu<<\mathcal{L}^2$. Set
\begin{equation}
\mathcal{G}\equiv\{A\subset\mathbb{R}^2\ |\ A\text{ is Borel and }\nu(\partial A)=0\}.
\end{equation}It is not hard to show that $\mathcal{G}$ is a $\sigma$-algebra on $\mathbb{R}^2$. Suppose that $\nu<<\mathcal{L}^2$ on $\mathcal{G}$ in the following sense: for all $\varepsilon>0$ there is an $\delta>0$ such that if $A\in \mathcal{G}$ and $|A|<\delta$, then $\nu(A)<\varepsilon$.

If $A$ is Borel but $\nu(\partial A)>0$, can $A$ be approximated by sets in $\mathcal{G}$? In particular, given $\varepsilon>0$, does there exist an $E\in\mathcal{G}$ such that $A\subset E$ and $\mathcal{L}^2(E\setminus A)<\varepsilon$?