Measurability of products If a set $\mathcal{N}\subseteq\mathbb{R}$ is not Lebesgue measurable, could it be possible that $\mathcal{N}\times\{0\}\subseteq\mathbb{R}^2$ is Lebesgue measurable? I do not think so since it is easy to construct a continuous function to show a contradiction to the positive answer. Can you guys give some comments or critics on my view? Many thanks!

$ \mathbb{R} × \lbrace 0 \rbrace $ in $ \mathbb{R^2} $ have measure of zero (null). $ \mathcal {N} × \lbrace 0 \rbrace \subseteq  \mathbb{R} × \lbrace 0 \rbrace $. then $ \mathcal {N} × \lbrace 0 \rbrace $ is measurable.
