I'm doing an exercise regarding a version of monotone class theorems for functions.
Let $\mathcal{C}$ be a collection of nonnegative bounded functions on $\Omega$. Then the following two statements are equivalent:
(1) $M(\mathcal{C})=\mathcal{L}_b^+(\mathcal{C})$
(2) $f,g\in\mathcal{C}\Rightarrow f\wedge g\in M(\mathcal{C})$; $f\in\mathcal{C},a\in \mathbb{R_+}\Rightarrow af, a-f\wedge a\in M(\mathcal{C})$
where $f\wedge g=\min\{f,g\}$.
$M(\mathcal{C})$ is the smallest collection of bounded functions on $\Omega$ containing $\mathcal{C}$, and the limit of any uniformly bounded monotonic sequence of functions in $M(\mathcal{C})$ is still in $M(\mathcal{C})$, i.e $f_n\in M(\mathcal{C}), |f_n|\leq C,\forall n\leq1, f_n\uparrow(\downarrow) f\Rightarrow f\in M(\mathcal{C})$.
$\mathcal{L}_b^+(\mathcal{C})$ is the collection of nonnegative bounded $\sigma(f|f\in\mathcal{C})$-measurable functions. $\sigma(f|f\in\mathcal{C})$ denotes the $\sigma$-algebra on $\Omega$ generated by functions in $\mathcal{C}$.
$\textbf{Sketch of what I have tried:}$
(1)$\Rightarrow$(2) is obvious, so we only need to prove $(2)\Rightarrow(1)$. Noting that $\mathcal{L}_b^+(\mathcal{C})$ is closed for every uniformly bounded monotonic sequence, we always have $\mathcal{L}_b^+(\mathcal{C})\supset M(\mathcal{C})$.
We only need to prove $\mathcal{L}_b^+(\mathcal{C})\subset M(\mathcal{C})$ when (2) holds.
Following a similar proof on my textbook, I think I can prove it in the following steps:
Step 1: Prove that $\forall f,g \in M(\mathcal{C}), a\in \mathbb{R_+}$, we have
$f\wedge g \in M(\mathcal{C}), af\in M(\mathcal{C}), a-f\wedge a \in M(\mathcal{C})$.
This can be prove with the argument used in the proof of monotone class theorem of sets.
Step 2: Prove that $\mathcal{F}=\{A\subset\Omega|I_A\in M(\mathcal{C})\}\supset\sigma(f|f\in\mathcal{C})$.
To prove this statement, one need to prove that $\mathcal{F}$ is a $\sigma$-algebra, and use the fact that $M(\mathcal{C})\ni f_n=[n(a-f\wedge a)]\wedge 1\uparrow I_{[f<a]}, \forall f\in \mathcal{C}, \forall a\in \mathbb{R}$, where $[f<a]=f^{-1}[(-\infty,a)]$.
Step 3: The last step is to approximate any $\sigma(f|f\in\mathcal{C})$-measurable functions with simple functions.
We have prove in step 2 that $\forall A\in \sigma(f|f\in\mathcal{C}), I_A\in M(\mathcal{C})$, but the problem is that we need the positive linear combination of functions in $M(\mathcal{C})$ to remain in $M(\mathcal{C})$ so that we can construct a sequence of simple functions in $M(\mathcal{C})$ which monotonically converges to a function in $\mathcal{L}_b^+(\mathcal{C})$.
I have no idea how to prove that $M(\mathcal{C})$ is closed under positive linear combination. Or should I modify the whole proof?
Any hint will be appreciated.
Best Answer
Note that $1-(1-(aI_A)\wedge1)\wedge(1-(bI_B)\wedge1)\wedge1=aI_A+bI_B$, if $A\cap B=\varnothing$ and $0\leq a,b \leq1$. So $M(\mathcal{C})$ is in fact closed under positive linear combination.