The proof of countable subadditivity of outer measure is given in pg.33 in Royden fourth edition as follows:
And the proof of theorem 18,pg.49, in the same edition is given by:
And this is the preceding theorem:
My question is:
I do not understand the proof of theorem 18, I want to prove it in details, specifically:
1- I do not understand how "by the very definition of a measurable set, every set must be measurable"?
here the definition of the measurable set we use:
So, could anyone provide me with a detailed proof of this theorem, please?
2- Will the proof similar to the above given for countable subadditivity? what is the difference in the proof and why?
Best Answer
You know that there are non-measurable set (for instance, Vitali set $\mathcal V$). If $m^*(A\cup B)=m^*(A)+m^*(B)$ for all disjoints set $A$ and $B$, then, in particular, $$m^*(A)=m^*(A\cap (\mathcal V\cup\mathcal V^c))$$ $$=m^*(A\cap \mathcal V\cup A\cap \mathcal V^c)\underset{(*)}{=}m^*(A\cap \mathcal V)+m^*(A\cap \mathcal V^c),$$ where $(*)$ come from the fact that $A\cap \mathcal V$ and $A\cap \mathcal V^c$ are disjoints. Thus $\mathcal V$ is measurable. Contradiction.