Let $E\subset \mathbb{R}$ be a set of positive Lebesgue measure. Assume that if $x,y\in E$ then $\frac{x+y}{2}\in E$. Prove that $E$ has at least one interior point.
Here is what I have done:
(1). By regularity, for any $\epsilon>0$ we can find an open set $O_\epsilon$ such that $E\subseteq O_\epsilon$ and $m(O_\epsilon)-m(E)<\epsilon.$ Write $O_\epsilon$ as a disjoint union of open intervals $\{I_j\}$ $$O_\epsilon=\bigsqcup_{j=1}^\infty I_j$$
(2). WLOG we can do the indexing in such a way that $I_{j+1}$ is the next interval to $I_j$ (in the sense that $I_{j+1}$ is on the right of $I_j$ and there is no $I_k$ which is in between $I_j$ and $I_{j+1}$.)
(3). If at least one $I_j\subseteq E$ then we are done. So assume that $I_j\subsetneq E$ for all $j$. Chose an $I_j$ and pick a point $x\in I_j\cap E$. Chose $y\in I_{j+1}\cap E$. Now $z=\frac{x+y}{2}\in E$ and thanks to the indexing, $z\in I_j$ or $z\in I_{j+1}.$ WLOG we can assume that $z\in I_j$.
(4) Now we have two point $x,z\in I_j$. We can recursively pick the midpoints on the line joining $x$ and $z$ and all these points will be in $E$. (First pick $\frac{x+z}{2}$, then pick $\frac{x+\frac{x+z}{2}}{2}$ and $\frac{z+\frac{x+z}{2}}{2}$ and so on)
(5). My guess is that one of the midpoints (constructed in the previous step) on the line joining $x$ and $z$ will be an interior point. But I don't know if my guess is correct.
Am I moving in the right direction? Is there a different way to solve this problem?
