Let $\mathcal{A}$ be the collection of all open sets in $\mathbb{R}^{n}$ and let $\mathcal{H}$ be the set of functions $f : \mathbb{R}^{n} \rightarrow \mathbb{R}$ such that
1) For any $A \in \mathcal{A}$, the function $I_{A}$ belongs to $\mathcal{H}$
2) $\mathcal{H}$ is a vector space
3) If $\{f_n\}_{n\in \mathbb{N}}$ is a sequence with $0 \leq f_n \leq f_{n+1}$ for every $n \in \mathbb{N}$, then $\lim_{n\to\infty} f_{n} \in \mathcal{H}$
Prove that $\mathcal{H}$ contains all Borel measurable functions.
This is a measure theory problem that I am having trouble with. I am trying to study for exams but I am really stuck on this problem. By the way, $I_{A}$ is the indicator set function. By vector space, I know that $\mathcal{H}$ has defined operations $+$ and $*$. I don't quite see how that one helps.
I'm thinking (3) can be used in some way sort of like squeeze theorem. But I am really clueless with this problem and I would like to understand it for my exam.
Thank you for any help
You mentioned in comments that you know the Dynkin $\pi$-$\lambda$ theorem, so here is an outline of how you could proceed. Let $\mathcal{P}$ be the collection of all open sets in $\mathbb{R}^n$, and let $\mathcal{L}$ be the collection of all sets $A \subset\mathbb{R}^n$ for which $I_A \in H$.
The significance of the vector space property is that whenever you know that two functions $f,g$ are in $H$, then you also know that $af+bg \in H$ for any scalars $a,b$. In particular, $f+g$ and $f-g$ are in $H$.
a. Observe that $\mathcal{P}$ is a $\pi$-system.
b. Show that $\mathcal{L}$ is a Dynkin system.
c. Conclude that $H$ contains $I_B$ for every Borel set $B$.
d. Show that $H$ contains every simple function (use the vector space assumption)
e. Show that $H$ contains every nonnegative Borel function (recall that every such function can be written as an increasing limit of simple functions)
f. Show that $H$ contains every Borel function (hint: $f = f^+ - f^-$).