Every subset of ${\mathbb{R}}^n$ has a $G_{\delta }$ measurable hull of the same outer measure

Given: A natural number $n\ge 1$, the Axiom of Countable Choice, and a subset $E\subseteq {\mathbb{R}}^n$.

Assuming countable choice, ${\lambda }_n^{\ast }(E)=\inf \{{\lambda }_n(U):U\subseteq {\mathbb{R}}^n$ open and $E\subseteq U\}$ for every subset $E$ (Assuming countable choice, the Lebesgue outer measure of an arbitrary subset of ${\mathbb{R}}^n$ is the infimum of the measures of the open sets containing it).

Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable (Assuming countable choice, every Borel subset of ${\mathbb{R}}^n$ is Lebesgue measurable).

Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra and ${\lambda }_n$ is a complete measure on it, and ${\lambda }_n$ is the restriction of ${\lambda }_n^{\ast }$ (Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume).

Assuming countable choice, ${\lambda }_n^{\ast }$ is an outer measure on ${\mathbb{R}}^n$, hence monotone (Assuming countable choice, Lebesgue outer measure is an outer measure that restricts to elementary volume, Outer measures, Lebesgue outer measure on ${\mathbb{R}}^n$).

$A$ is a $G_{\delta }$ set of $X$ when there is a sequence $(V_n{)}_{n\in \mathbb{N}}$ of open subsets of $X$ with $A={\bigcap }_{n\in \mathbb{N}}{V}_n$ ($G_{\delta }$ and $F_{\sigma }$ subsets of a topological space, agreeing with the real-line notion).

A measurable hull of $E$ is a Carathéodory measurable set $H\supseteq E$ with ${\mu }^{\ast }(H)={\mu }^{\ast }(E)$; the outer measure is regular when every subset has a measurable hull (Measurable hulls and regular outer measures).

Assume the Axiom of Countable Choice. An outer measure induced by a premeasure is regular, and every set has a measurable hull in $\sigma ({\mathcal{A}}_0)$ (Assuming countable choice, a premeasure-induced outer measure is regular with generated measurable hulls).

The product topology on ${\mathbb{R}}^n$ is the metric topology of $d_2$ (A subset of ${\mathbb{R}}^n$ with the product topology is compact exactly when it is closed and bounded, the product topology being the Euclidean metric topology, claim 1; The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

For every real $\epsilon >0$ there is a natural number $k\ge 1$ with $1\mathrm{/}k<\epsilon$ (For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).

The Axiom of Countable Choice says that for every family $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets indexed by $\mathbb{N}$ there is a function $f$ with domain $\mathbb{N}$ such that $f(n)\in X_n$ for every $n\in \mathbb{N}$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).