Consider a fixed rotation of the usual coordinate system in $\mathbb R^n$. Notions pertaining to the rotated system will be denoted by primes. Thus, $I'$ denotes an interval with edges parallel to the rotated coordinate axes, and $|E|'_e$ denotes the outer measure of a subset $E$ relative to these rotated intervals: $$|E|'_e = \inf \sum v(I'_k)$$ where the infimum is taken over all coverings of $E$ by rotated intervals $I'_k$.
In the proof, they first claim that given $I^{'}$ and $\epsilon > 0$, there exist $\{I_l\}$ such that $I^{'} \subset \cup I_l$, and $\sum v(I_l) \le v(I^{'})+\epsilon$. Then, they show that the claim is true when $I^{'} \subset I_1^{'}$, and they say the claim is hence proven. How does this case help to prove the claim? I think the parallel result is exactly the same as their claim at the beginning. Why do they state it again? On $-2$ line, should $E \subset \cup_{k,l} I_{k,l}^{'}$ be $E^{'} \subset \cup_{k,l} I_{k,l}^{'}$?
