Currently I'm just working through measure theory just to see if its something I would like to take.
Unfortunately I am stuck on this problem from Carothers.
If $m^*(E)=0$, then $m^*(E^2)=0$.
Where $m^*$ denotes outer measure and $E^2=\{x^2:x\in E\}.$
I toyed with the idea that $I_k < 1 \Rightarrow I^2_k < I_k$. However I am at a loss as to how to set up a chain of inequalities (which is what I am assuming I need).