TITLE: Projection of measure zero set in $\mathbb{R}^2$.
QUESTION [1 upvotes]: 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 $\mathbb{R}$?

REPLY [3 votes]: No. Consider the set $E=\{(x,y):x=y\}$. This set has measure $0$ but its projections are measurable and do not.
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