Let $A,B\subset \mathbb{R}$ be such that they are positive Lebesgue measure in $\mathbb{R}$. Let $C\subset \mathbb{R}^2$ is closed, nowhere dense, measure zero set in $\mathbb{R}^2$.
Does there exists $E,F \subset \mathbb{R}$ satisfying $$E×F\subset A×B\setminus C,$$ such that $E,F$ are positive Lebesgue measure in $\mathbb{R}?$
Edit: I worked out the problem (though I am not sure whether it is correct or not, that's why I am editing the question instead of posting in answer section)
$\textbf{ My try:}$ It is enough to assume that $C$ is closed and measure zero set in $\mathbb{R}^2$. So $A×B\setminus C$ is an open set in $A×B$ and hence there exists a sequence of almost disjoint ( i.e. intersection of their interior are disjoint) closed squares $\{A_k\}$ in $\mathbb{R}^2$ such that $$A×B\setminus C= \cup_{k\in \mathbb{N}}A_k\cap A×B.$$ Since $A×B\setminus C$ is again a set of positive measure and $A_k$'s are almost disjoint we must get $m\in \mathbb{N}$ such that $A_m\cap A×B$ is positive measure. Now let us consider
$$E=\{ \text { projection of }A_m \text{ onto } x\text{-axis}\}\cap A,$$
$$F=\{ \text { projection of }A_m \text{ onto } y\text{-axis}\}\cap B.$$
Then $E×F=A_m\cap A×B$ and $E,F$ are positive measure in $\mathbb{R}$ as $A_m\cap A×B$ is positive measure in $\mathbb{R}^2.$