Structure of Measurable Subsets of the Unit Square - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:53:09Z http://mathoverflow.net/feeds/question/83536 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83536/structure-of-measurable-subsets-of-the-unit-square Structure of Measurable Subsets of the Unit Square Stephan 2011-12-15T17:50:11Z 2011-12-15T18:05:46Z <p>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?</p> http://mathoverflow.net/questions/83536/structure-of-measurable-subsets-of-the-unit-square/83539#83539 Answer by Robert Israel for Structure of Measurable Subsets of the Unit Square Robert Israel 2011-12-15T18:05:46Z 2011-12-15T18:05:46Z <p>Consider a measurable subset $S$ of $I = [0,1]$ with positive measure. Then <code>$A = \{(x,y) \in I^2: x - y \in S\}$</code> has positive measure. Suppose $B_1$ and $B_2$ have positive measure. Then it is well-known that <code>$B_1 - B_2 = \{ x - y:\ x \in B_1,\ y \in B_2 \}$</code> contains an interval of positive length. So if $S$ contains no such interval, $B_1 \times B_2$ can't be a subset of $A$.</p>