The Lemma and part of its proof are given below:
My questions are:
1- What is the relation between that the intersection of a countable collection of measurable sets is measurable and $E_{n}$ being measurable, $E_{n}$ is the intersection of what sets?
2- Why $\{ E_{n} \}$ is an ascending collection of measurable sets?
Could anyone help me in answering this questions please?
Each $E_n$ can be written as $$E_n=\bigcap\limits_{k=n}^\infty\{x\in E: |f(x)-f_k(x)|<\eta\}.$$ The inner set is measurable, as justified by Royden, and the countable intersection of measurable sets is measurable, so each $E_n$ is measurable. As for why the sets are ascending, just note that as $n$ grows, we're requiring the bound for "less" functions, so more $x$'s will be admissible. Explicitly, if $x\in E_n,$ then $$|f(x)-f_k(x)|<\eta$$ for all $k\geq n.$ If $m>n,$ then this is also true for all $k\geq m,$ so $x\in E_m,$ too.