Prove function is measurable

Question

I was watching this exercise

Let $\{a_{ij}\}\subseteq [0,+\infty]$ be a sequence. Prove that $$\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}a_{ij} = \sum_{j=0}^{\infty}\sum_{i=0}^{\infty}a_{ij}$$

If we define $f_n(i) = \sum\limits_{j=0}^n a_{ij}$. Is easy to see that is a monotone sequence and the idea is use the monotone convergence theorem, but I have problems to prove that $f_n$ is measurable with the counting measure defined as $\mu: {\cal P}(\mathbb{N})\rightarrow[0,+\infty]$.

Actually I have problems to prove if a function is measurable in general, some advise or paper to learn?

Related questions: Positivity and Interchange of Summation, Under what condition we can interchange order of a limit and a summation? and When can you switch the order of limits? — Martin Sleziak, Oct 20 '13 at 17:10

score 2 · Answer 1 · answered Oct 20 '13 at 16:30

2

A function $f:X\to [0,\infty]$ is measurable iff $f^{-1}(V)$ is measurable for any open set $V \subset [0,\infty]$. In this case, $f^{-1}(V) \in \mathcal{P}(\mathbb{N})$, so it is measurable.

answered Oct 20 '13 at 16:30

Prahlad Vaidyanathan

31,554

Prove function is measurable

1 Answers1