I was trying to prove this statement:
Prove that every measurable set $E \subset \mathbb{R}$ is the union of a Borel set and a set of measure 0.
And I found its prove here Prove that Lebesgue measurable set is the union of a Borel measurable set and a set of Lebesgue measure zero
But the proof in the link is unclear for me (seems to be not organized) and it uses the first part in the question in the link, So I am wondering is there a nice and easier proof than this? I am studying measure from Royden "Real Analysis" and I have read until theorem 12 on section 2.4. Any help in an organized simple proof will be greatly appreciated.
EDIT:
I have taken this theorem:
So what can I do after using (iii) in this theorem as a given?