So the statement for Monotone Class Theorem goes:
Let $(\Omega,\mathcal{F})$ be a space with a $\sigma$-algebra $\mathcal{F}$ and with $\pi$-system $\mathcal{A}$ generating $\mathcal{F}$. If $ V $ is the $\mathbb{R}$-vector space of functions satisfying:
1) $\mathbb{1}_{A} \in V$ for all $A \in \mathcal{A}$ and $\mathbb{1} \in \mathcal{A}$,
2) for any $0 \leq f_n \in V$ with $f_n \uparrow f$ pointwise, then $f \in V$,
then $V$ contains all bounded $\mathcal{F}$-measurable functions.
So the proof uses Dynkin's $\pi$-system lemma to show we get all indicator functions of sets in $\mathcal{F}$, then use the upwards approximation by simple functions of the form $2^{-n}\lfloor2^{n}f\rfloor \wedge n$.
My question is whether we can extend the result to include all measurable functions, including unbounded ones. It seems like the proof carries over, and one might think that boundedness is necessary to ensure the construction $2^{-n}\lfloor2^{n}f\rfloor \wedge n$ takes on finitely many values but it already does so as we are bounding it by $n$.
Is there a subtlety that I may be missing? Like say, we are allowing the function to take the value $\infty$ at certain points?