Result about Lebesgue measurable subset of $\mathbb{R}^2$ and horizontal/vertical slices

Question

I am studying for my qualifying exam in Real Analysis (Measure Theory) and am stuck on the following practice problem:

Let $E$ be a Lebesgue measurable subset of $\mathbb{R}^2$. Suppose that for every $x \in \mathbb{R}$, the vertical slice $E_x = \{ y \in \mathbb{R} : (x,y) \in E \}$ has positive Lebesgue measure. Prove that there is a subset $A$ of $\mathbb{R}$ of positive Lebesgue measure such that for all $y \in A$, the horizontal slice $E^y = \{ x \in \mathbb{R} : (x,y) \in E \}$ is uncountable.

I first tried going about it by contradiction, but couldn't get that to work. Then I tried letting $A = \bigcup E_x$, but that didn't get me anywhere either. Any help is greatly appreciated!

Answer 1

Hint: Fubini's theorem applied to the indicator function of $E$.