Lipschitz solvability of prescribed Jacobian and divergence for singular measures

Let $\mu$ be a finite Radon measure on an open set $\Omega\subset\mathbb{R}^d$, singular with respect to the Lebesgue measure. We prove Lusin-type solvability results for the prescribed divergence equation and the prescribed Jacobian equation with Lipschitz solutions. More precisely, for every $\varepsilon>0$ and every Borel datum $f \colon \Omega \to \mathbb{R}$ there exists a vector field $V\in C^1_c(\Omega;\mathbb{R}^d)$ such that $\operatorname{div} V=f$ on a compact set $K\subset\Omega$ with $\mu(\Omega\setminus K)<\varepsilon$, and $\operatorname{Lip}(V)\le (1+\varepsilon)\|f\|_{L^\infty(\Omega,\mu)}$. Similarly, for every Borel datum $g\colon \Omega \to \mathbb{R}$ there exists a map $\Phi$ with $\Phi-\operatorname{Id}\in C^1_c(\Omega;\mathbb{R}^d)$ such that $\det D\Phi=g$ on a compact set $K\subset\Omega$ with $\mu(\Omega\setminus K)<\varepsilon$, and $\operatorname{Lip}(\Phi-\operatorname{Id})\le (1+\varepsilon)\|g-1\|_{L^\infty(\Omega,\mu)}$. The maps $V$ and $\Phi-\operatorname{Id}$ can be chosen arbitrarily small in supremum norm.