Verification of argument in Corollary 3.3 Stein and Shakarchi Real analysis

Question

I thoght Proof of Corollary 3.3 (ii) is not complete.

As E is not shown to be measurable there.

I thought following argument . As the complement of a measurable set is measurable then $E_1^c, E_2^c....$ are measurable.

this sequence is increasing like (i)

Using that $m(E^c)=\lim_{N\to\infty}m(E_N^c)$

and $E^c=\cup E^c_N$ so it is measurable.

Complement is also measurable

SO E is measurable.

Please See my argument. If there is any other argument please suggest.

Answer 1

$E$ is shown to be a measurable set here; from the marked identity:

$$E = E_1 \setminus \bigcup_{k=1}^\infty G_i$$

$E_1$ is measurable and so are the $G_i$ as differences of measurable sets so $E$ is measurable. I don't see any incompleteness. The author could have made it more explicit perhaps. But from the start $E=\bigcap E_n$ is clearly measurable when all $E_n$ are. (That's what $E_n \searrow E$ means.)