Do there exist any subsets of $\mathbb{R}$ with positive measure but not of size continuum?

Question

As title goes, without assuming continuum hypothesis, is there a subset of $\mathbb{R}$ with positive measure but not being continuum? That is,

Does there exist $A\subseteq \mathbb{R}$ such that $\mu(A)>0$ and $\aleph_0<|A|<2^{\aleph_0}$?

I believe that maybe in some system without assuming continuum hypothesis, there exists.

And by the regularity of Lebesgue measure, it suffices to deal with compact set.

Answer 1

There's a standard exercise in measure theory that says that if $A$ has positive measure, then $A-A$ contains an interval. It follows from this that $A$ must have the same cardinality as the reals.

Answer 2

I find a solution due to This question. Due to the regularity of Lebesgue measure, it suffices to deal with compact set. And a standard result is a closed subset of $\mathbb{R}$ is either at most countable or continuum.