there exists two sets $A,B$ such that $A \times B \subset E$ and $0<m_{1}(A)m_{1}(B)$ for $m_{2}(E)=1$

Question

Suppose $E \in [0,2] \times [0,2]$ and $m_{2}(E)=1$ where $m_{2}$ is the two-dimensional Lebesgue measure. Show that there exists two sets $A,B$ such that $A \times B \subset E$ and $0<m_{1}(A)m_{1}(B)$ where $m_{1}$ is the one-dimensional Lebesgue measure.

I tried to use Fubini theorem, but I failed to use it.

p.s. I guess that the above statement is false.

Answer 1

Let $E$ be the intersection of $[0,2]\times[0,2]$ with the complement of the union of all lines of slope $1$ that go through points with rational coordinates.