Uncountable dense measurable subgroups of $\mathbb{C}$

Question

Is it possible to have an uncountable proper dense subgroup of $\mathbb{C}$ which is also Baire or Lebesgue measurable?

Answer 1

It seems the following.

Lebesgue measurable

As I already noted, the group $\mathbb Q\times\mathbb R$ is a required.

Baire

The construction of such a group $G$ requires Axiom of Choice, because $G$ is descriptively complex. I recall the following definitions. A subset $A$ of a topological space $X$ is meager in $X$, if $A$ is contained in the countable union of nowhere dense subsets of $X$. A subset $B$ of a topological space $X$ has the Baire Property in $X$ if $B$ contains a $G_\delta$-subset $C$ of $X$ such that $B\setminus C$ is meager in $X$. By [Kech, 8.22] each Borel subset of a space $X$ has the Baire Property in $X$. By [Kech 9.8], a topological group $G$ is Baire iff it is non-meager (in $G$). It is easy to check (or see Proposition 1.3 or the beginning of section I.2 from [HC] ) that a dense subspace $Y$ of a topological space $X$ is meager in $X$ iff $Y$ is meager in $Y$. At last, by Banach-Kuratowski-Pettis Theorem (see [Kel, p.279] or [Kech 9.9]), for any subset $B$ of a topological group the set $BB^{-1}$ is a neighborhood of the unit provided $B$ is non-meager and has the Baire Property in the group. So, such a proper dense Baire subgroup of $\mathbb C$ cannot be Borel or, even, cannot have the Baire Property in $\mathbb C$.

For an example of such a group $G$ consider the group $\mathbb C$ as a vector space over the field $\mathbb Q$ and let $G$ be any vector subspace of $\mathbb C$ of codimension $1$. Since $\mathbb C$ is a countable union of copies of $G$, the group $G$ is non-meager in $\mathbb C$. Using this fact it is easy to show that $G$ is dense in $\mathbb C$. So the group $G$ is non-meager in $G$, therefore $G$ is Baire.

References

[HC] R. C. Haworth, R. C. McCoy, Baire spaces, Warszawa, Panstwowe Wydawnictwo Naukowe, 1977.

[Kech] A. Kechris. Classical Descriptive Set Theory, – Springer, 1995.

[Kel] J. Kelley. General Topology. -- M.: Nauka, 1968 (in Russian).