Product of Borel and non-Borel set

Question

It is true that product of Borel and non-Borel sets is non-Borel set? More precisely, I would like to know if $V $×$ \{1\}$ is Borel, where $V$ is Vitali set.

Answer 1

It is well known that for a Borel set $M \subset \Bbb{R}^n \times \Bbb{R}^m$ the "sections"

$$ M_x =\{y\in \Bbb{R}^m \mid (x,y)\in M\},\\ M^y =\{x\in \Bbb{R}^n\mid (x,y)\in M\} $$ are Borel measurable for all $x\in \Bbb{R}^n,y\in \Bbb{R}^m$. To see this, show that the class of all such sets forms a sigma algebra an contains the open sets.

Using this property, it is easy to see that if $M,N$ are nonempty and if $M\times N$ is Borel, then so are $M,N$.