I have a few questions in my mind:

1) [tex] \lim_{n\rightarrow \infty}[0,n) = \cup_{n\in\mathbb{N}}[0,n) = [0,infty) [/tex] holds, and for [tex] \lim_{n\rightarrow \infty}[0,n] = \cup_{n\in\mathbb{N}}[0,n] = [0,infty) [/tex] is also true? It should not be [tex] [0,infty] [/tex], am I correct?

2) Consider an extended real function [tex] f [/tex], if we use simple function [tex] f_{n} = f 1_{f\leq n} [/tex], by taking limit, we can only have it approximated to [tex] f 1_{f < \infty} [/tex] but since [tex] f [/tex] may take [tex] \infty [/tex], such simple function may not be approximating [tex] f [/tex] almost everywhere unless [tex] f = \infty [/tex] is of measure 0?

Am I correct? Thanks.

Wayne