Let $A\subset\mathbb{R}$ be a nonmeasurable set, and let $E\subseteq\mathbb{R}$ be a (Lebesgue) measurable set.
How do we show that the Cartesian product $A\times E$ is nonmeasurable in $\mathbb{R}^2$?
Is there any way to show it without referring to the concept of product measure? I can understand (How to show product of two nonmeasurable sets is nonmeasurable?) but would like a more concrete method. Perhaps using Caratheodory criterion or something like that?
Thanks for any help. Hope the question makes sense..