The proposition and its proof are given below:
My questions are:
1-I do not understand the proof in the first line and half, why since each interval $(c, \infty)$ is open the function $f$ is measurable?
2- why $\mathcal{O}$ can be written as the union of bounded intervals? and are those intervals disjoint? if so, why we want them disjoint? if not, why we do not want them disjoint ?
Could anyone help me in answering this question, please?
It is an ''if and only if'' statement,so in the first line he assumes that the inverse of an open set is measurable.
It is a fact that on the real line every open set can be written as a countable union of open disjoint intervals $I_n$
You can make these intervals bounded considering the intervals $I_{m,n}=I_n \cap (-m,m)$ where $m,n \in \Bbb{N}$.
In general it does not matter if we take the intervals whose union is an open set disjoint or not for this proof.