Given a compact set $A$ in $\mathbb{R}^2$ with positive 2-dimensional Lebesgue measure, does there always exist $S, T \subset \mathbb{R}$ both Lebesgue measurable and non empty, such that $\mu (S)>0$ or $\mu (T) > 0$, and $S \times T \subset A$ ?
I found answers to similar questions, but the hypothesis were weaker : $A$ Borel measurable, or $S$ and $T$ both non null sets (and with such hypothesis, the statement is false).
I also found a theorem from Mycielski which might answer this question. The reference is
J. Mycielski, Algebraic independence and measure, Fund. Math. 61 (1967), 165-169
but I could not find it. Apparently, the theorem states that :
Suppose $A$ is a compact subset of plane of positive area. Then there are non empty perfect sets of reals $X$,$Y$ such that $X$ or $Y$ has positive length and $X \times Y \subset A$.
But I think that I am lacking context. Any help on the problem itself, or a link to a page where I could find the complete statement of the theorem, would be greatly appreciated.