A Lebesgue measurable subgroup of $({\mathbb{R}}^n,+)$ of positive measure is all of ${\mathbb{R}}^n$

Given: A natural number $n\ge 1$, the Axiom of Countable Choice, and a Lebesgue measurable subgroup $G$ of $({\mathbb{R}}^n,+)$ with ${\lambda }_n(G)>0$.

Assuming countable choice, a Lebesgue measurable $E\subseteq {\mathbb{R}}^n$ with ${\lambda }_n(E)>0$ has a real $r>0$ with $B(0,r)\subseteq E-E$ (If a Lebesgue measurable subset of ${\mathbb{R}}^n$ has positive measure, its difference set contains an open ball about the origin, Open ball, closed ball and sphere in a metric space).

Assuming countable choice, ${\lambda }_n$ is a complete measure on the sigma-algebra $\mathcal{L}({\mathbb{R}}^n)$ (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).

A subset $H\subseteq G$ is a subgroup when $e\in H$, $H$ is closed under the operation, and $H$ is closed under inverses (Subgroup, Group and abelian group).

Every complete ordered field $F$ is Archimedean: for every $x\in F$ there is a natural number $m\ge 1$ with $x<m\cdot 1_F$ (Every complete ordered field is Archimedean); and for every real $\epsilon >0$ there is a natural $k\ge 1$ with $1\mathrm{/}k<\epsilon$ (For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).

Let $S\subseteq \mathbb{N}$; if $0\in S$ and $\sigma (m)\in S$ whenever $m\in S$, then $S=\mathbb{N}$ (The principle of mathematical induction).

$d_2(x,y)=\parallel x-y{\parallel }_2$, and $\parallel \lambda x{\parallel }_2=\mathrm{\mid }\lambda \mathrm{\mid }\parallel x{\parallel }_2$ for a real $\lambda$ (Each $\parallel \cdot {\parallel }_p$ is a norm on ${\mathbb{R}}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_{\mathrm{\infty }}$ of the published metric-spaces page, claims 1 and 3; The $p$-norms $\parallel x{\parallel }_p$ for rational $p\ge 1$, and $\parallel x{\parallel }_{\mathrm{\infty }}$, ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it).