So here are the questions came in my mind, but I could not answer each of them..
If $A$ and $B$ are two subsets of $\Bbb{R}$ then $\lambda_2^*(A×B)=\lambda_1^*(A) \lambda_1^*(B)$?? (If not the case, then what types of counter we should look for?) where $\lambda_2^*(A)$ is Lebesgue outer measure of any subset $A$ in $\Bbb{R}^2$ and $\lambda_1^*(A)$ is Lebesgue outer measure of any subset $A$ in $\Bbb{R}$. (I can only prove that $\lambda_2^*(A×B) \leq \lambda_1^*(A) \lambda_1^*(B)$.)
And now if the above does not hold always, then in what condition or restriction we should give two the sets $A$ and $B$ such that the above equality hold, and how to prove it?? (Not just $A$ and $B$ are measurable; then it holds. I want to look for some general fact.)
Any hint will be appreciated.