Plane measurable sets and measurable rectangle 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?
 A: No. 
Let $K$ be a Cantor set in the unit interval of positive one-dimensional Lebesgue measure. More generally, we can take any measurable subset of $\bf R$ of positive Lebesgue measure and empty interior. Rotate $K\times K \subset {\bf R}^2$ by a quarter turn (45 degree) in the plane.  The resulting set cannot contain a subset $A\times B$ with $A$ and $B$ of positive measure.
This is shown as follows. Let us project our set on the line of slope -1 through the origin, graduated so that the point $(x,y)$ is sent to the point $x-y$ on the line.
It is a standard fact that if $A$ and $B$ are two subsets in $\bf R$ of positive Lebesgue measure, then the set $A-B=\{x-y \mid x\in A, \ y\in B\}$ contains an interval. This follows from the continuity of $x\mapsto \int {\bf 1}_A(t-x) {\bf 1}_B(t) \ dt$.
So the projection of our set on the line must contain an interval. But this projection is (a translation-rotation of) $K$ which is of empty interior.
A: 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.
