Elaborating some points in the proof of the countable additivity of integration.

Question

The statement and its proof are given below:

My questions are:

1- why $\{f_{n}\} \rightarrow f$ pointwise on $E$?

2- In the third line from below:

Should it be $\int_{E_{k}} f_{n}$ instead of $\int_{E_{k}} f$, as in the statement of "the additivity over domains of integration" there is only one function not two as you can see from below. could anyone explains this for me , please?

Answer 1

For 2, it is correct because $f_n=fχ_{\bigcup_{k=1}^nE_k}=f \sum_{k=1}^n1_{E_k}$ since $E_k$ are disjoint.

For 1.

$1_{\bigcup_{k=1}^nE_k} \to 1_E$ pointwise thus $f_n \to f$ pointwise.