Projection of measure zero set in $\mathbb{R}^2$.

Question

Let $E\subset \mathbb{R}^2$ be a set of measure zero in $\mathbb{R}^2$. Let $E_x,E_y$ be the projection of E on the $x$-axis and $y$-axis respectively.

Is it true that atleast one of $E_x$ and $E_y$ is measurable and measure zero in $\mathbb{R}$?

Answer 1

No. Consider the set $E=\{(x,y):x=y\}$. This set has measure $0$ but its projections are measurable and do not.