Can one construct a set $E$, $m^*(rE)\neq rm^*(E)$.
Well , we know that if $E$ is measurable then $$m^*(rE)= rm^*(E)$$,so what is the case of a none-measurable set ?
Well,I think it holds for any set $E$, we assume that $$m^*(E)=\sum_n |I_n|$$ Where $I_n$ are some cubes which covers the $E$ , since for all $x\in E$, there will exist $x\in I_k$, thus $rx\in rE,rx\in rI_k$ , which implies $\{rI_n\}$ is an open cover of $rE$,thus $$m^*(rE)\leq rm^*(E)$$ On the other hand , assume $\{J_n\}$ is an open cover of $rE$ , one can use $1/r J_n$ to gain the reverse inequality.
I don't know whether there are any bugs in my proof , so , hope your nice answer, thank you in advance