Problem: As I've mentioned before, I am studying Probability and Measure Theory on my own, and I am using Resnick as the main text. I've been working through Ch. 3 on random variables, measurable maps, etc. and am somewhat stuck on one of the exercises.
Suppose we have a space $\Omega$ and a countable partition: {$B_n, n\geq 1$} of that space. Next, we define the sigma field $\mathcal{B}=\sigma(B_n,n\geq 1)$.
We are asked to show that a function $X:\Omega\rightarrow (-\infty,\infty]$ is $\mathcal{B}$-measurable iff it is of the form:
$\sum_{i=1}^\infty c_i 1_{B_i}$, for constants {$c_i$}.
Understanding/Attempt at a solution:
If I understand the problem (a large, but variable "if"), we need to prove that a random variable is measurable iff it can be expressed as a simple function, because simple functions are measurable. So, in effect, the random variable can only take values that correspond to some combination of disjoint sets that partition the space?
In my attempt, I start with $B_i \in \mathcal{B}$ and show that $X(\omega)=1_{B_i}(\omega)$ is measurable because $\varnothing, B_i^c, \Omega \in \mathcal{B}$.
I am not sure what to do next, although it seems like I should be trying to show that the inverse image of (c,$\infty$] under X should be a union of elements of $\mathcal{B}$.
Even then, it only feels like I am addressing the "if" part, and I'm not sure what to do with the "only if" part of the exercise.
As always, thank you for any help you can provide!