Is there a set $A \subset [0,1]$ such that both $A$ and $[0,1] \setminus A$ intersect every positive-measure set?

Is there a set $A \subset [0,1]$ such that for every Borel set $B \subset [0,1]$ of positive Lebesgue measure, both $B \cap A$ and $B \setminus A$ are non-empty?

This is, in a sense, the "measure-theoretic analogue" of the more obvious topological question: is there a set $A$ such that for every non-empty open $U \subset [0,1]$, both $U \cap A$ and $U \setminus A$ are non-empty? (For which the answer is obviously $A:=\mathbb{Q}$.)

Now it is clear that $A$ cannot itself be a Borel set of non-trivial Lebesgue measure (just take $B=A$). My intuition is that $A$ cannot be any Lebesgue-measurable set.

I thought about taking $A$ to be a set consisting of one point from every orbit of $x \mapsto x+\sqrt{2} \; \mathrm{mod} \; 1$, or at least a union of such sets. But I'm not sure if this gets anywhere.

• transfinite recursion.... – Henno Brandsma Mar 11 '17 at 16:08

As shown in this question there are $\mathfrak c$ Borel sets, which can be extended to the fact that there are $\mathfrak c$ Borel sets of positive measure. Each one with positive measure must have continuum many points. Put them in a well order in type $\mathfrak c$, so each one has $\lt \mathfrak c$ predecessors. Take each set in turn and find two points that have not been accounted for yet. Put one in $A$ and one in $\overline A$. As by the time you get to each set you have accounted for less than $\mathfrak c$ points, you can always find two new points to work with. If you have any points not accounted for when you are done with the positive Borel sets, put them wherever you want.
• This in fact proves that $A$ can be taken such that $A$ and $[0,1] \setminus A$ intersect every Borel set of size $\mathfrak{c}$. – Julian Newman Mar 11 '17 at 18:42
• If you only run this construction for perfect sets (rather than Borel sets of positive measure) you get what is known as a Bernstein set. It's a very similar beast, and it is enough to answer the question. In fact, a Bernstein set meets every Borel set of size $\frak c$, not just those of positive measure. This is because every Borel set is countable, or has a perfect subset. – Asaf Karagila Mar 13 '17 at 7:53