I came up to this question while using the Lusin's theorem, anyway the question goes as follows:
Let $X,Y$ be compact topological spaces endowed with the Borel $\sigma$-algebra and Borel measures $\mu_X,\mu_Y$ (with total measure $1$). I consider the product space $X\times Y$ with the standard product $\sigma$-algebra (which is the Borel $\sigma$-algebra of the product space) and the product measure.
Suppose that there exists a Borel set $E\subseteq X\times Y$ of large measure $\mu_X\times\mu_Y (E)>1-\varepsilon$. Such that property $P$ is satisfied for all $(x,y)\in E$.
Can I deduce that:
There exists sets $E_X,E_Y$ of large measure $\mu_X(E_X),\mu_Y(E_Y)>1-o(\varepsilon)$ such that property $P$ is satisfied for all $y\in E_Y$ and all $x\in E_X$?
In other words given a set of large measure in the product space $E\subseteq X\times Y$, can you find a product set $E_X\times E_Y$ that is also of large measure in every coordinate?
Let $\mu_X =\mu_Y=$ Lebesgue measure on $[0,1]$ and $ \mu = \mu_X \times \mu_Y$. Then $\mu \{(x,y):x=y\}=0$. Hence $\mu \{(x,y):|x-y| <\delta \} \to 0$ as $\delta \to 0$. Let $E$ be the complement of $\{(x,y):|x-y| <\delta \}$ and choose $\delta$ such that $\mu (E) >1-\epsilon$. Suppose there are sets $A$ , $B$ in $[0,1]$ such that $x \in A$ and $y \in B$ implies $(x,y) \in E$ and $\mu_X(A) >\frac 1 2,\mu_Y(B) >\frac 1 2$. Then $A$ and $B$ are disjoint so $1=\mu_X [0,1]\geq \mu_X(A)+\mu_X (B) >\frac 1 2 +\frac 1 2 =1$, a contradiction.