Question

If A is a (Lebesgue-)measurable subset of the unit square that has positive measure, does there exist a subset B contained in A that has a product structure (is the product of two subsets of the real line) and that has positive measure?

Answer 1

Consider a measurable subset $S$ of $I = [0,1]$ with positive measure. Then $A = \{(x,y) \in I^2: x - y \in S\}$ has positive measure. Suppose $B_1$ and $B_2$ have positive measure. Then it is well-known that $B_1 - B_2 = \{ x - y:\ x \in B_1,\ y \in B_2 \}$ contains an interval of positive length. So if $S$ contains no such interval, $B_1 \times B_2$ can't be a subset of $A$.