TITLE: Plane measurable sets and measurable rectangle
QUESTION [5 upvotes]: Does every measurable set in the plane with positive Lebesgue measure contain a cartesian product of two measurable sets of the real line with positive Lebesgue measures?

REPLY [3 votes]: Here are some variations on your question:
Fact: (Mycielski) Suppose $A$ is a compact subset of plane of positive area. Then there are perfect sets of reals $X, Y$ such that $X$ has positive length and $X \times Y$ is contained in $A$.
Question: Suppose $A$ is a subset of plane of positive area. Must $A$ contain a non null rectangle (a rectangle is a set of the form $X \times Y$)?
Answer: No.
Question: Suppose $A$ is a subset of plane of zero area. Must the complement of $A$ contain a non null rectangle?
Answer: (H. Friedman and (independently) Shelah) Independent of ZFC. True under CH but false in Cohen's model for the negation of CH.
Question: Suppose $A$ is dense $G_{\delta}$ subset of plane. Must $A$ contain a non meager rectangle?
Remark: Yes under CH but I don't know if this is true in ZFC alone.