Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer 1

up vote 5 down vote accepted

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 $A$.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.