Doubt in a proof of Steinhaus Theorem

Question

I am reading an elementary proof of Steinhaus theorem from a wiki page:

https://en.wikipedia.org/wiki/Steinhaus_theorem#Proof

I cannot see the step: "For our purpose it is enough to choose $K$ and $U$ such that ${\displaystyle 2\mu (K)>\mu (U).}$"

Why can we choose such sets? I have tried to use Lebesgue's regularity but I don't get those sets.

Any kind of help is thanked in advanced.

Answer 1

Take $\epsilon <\frac {\mu (A)} 4$. Then the condition $\mu (K)+\epsilon >\mu (U) -\epsilon$ implies that $2\mu (K) >\mu (U)$. [ $2\mu(K) >2 \mu(U) -4\epsilon >\mu (U)$ (because $\mu (U) \geq \mu (A)$).