I have a zero measure set $Z \subseteq \mathbb{R}$ and a non-measurable set $N \subset \mathbb{R}$. I want to conclude that $\lambda(Z \times N)=0$. I know that $$ Z \times N \subset Z \times \mathbb{R} $$ and $\lambda(Z \times \mathbb{R}) = \lambda(Z) \lambda(\mathbb{R}) = 0 \cdot \infty = 0$ so $\lambda(Z \times N) \leq \lambda(Z \times \mathbb{R}) =0$, which begs the question of measurability of $Z \times N$. Are we to conclude that it is measurable because the Lebesgue measure is complete.
Kindly help me out.
Yes, this set is mesurable and it has measure 0. One of the properties of Lebesgue measure is as follows:
If $A \subseteq B$, $B$ is measurable and $\lambda(B)=0$, then $A$ is measurable and $\lambda(A)=0$.
This is exactly what you need here.