Let $A\subset\mathbb{R}^p$ and $B\subset\mathbb{R}^q$, it’s not difficult to show that $$m^*(A\times B)\leq m^*(A)\cdot m^*(B)$$, where $m^*()$stands for the outter measure in Lebesgue meaning.

If A and B are measurable, then "=" holds. My question is whether "=" holds for all of/ none of/ some of the non-measurable A and B?

Construction of A and B in these different cases is needed.