I have a indirect question about Monotone Class Theorem (MCT), in its functional form. Here is a version which should be sufficiently general for my purpose.
Functional Monotone Class Theorem :
Let H a collection of real valued and bounded functions such that:
- H is a vector space
- H contains constants
- If $f_n$ is an increasing sequence of real and bounded functions of H, admitting a limit $f$ that is a bounded real valued function, then $f$ belongs to H.
Now let K be another collection of real bounded functions which is stable by multiplication, and such that $K \subset H$.
Then H contains all real bounded functions that belongs to $\sigma(K)$ (the sigma algebra generated by K).
The statement here is clear but I have a question about applications of it.
Usually the collection K is built by considering a collection P over real valued functions satisfying some propetry and intersecting it with the collections of bounded functions. Then using MCT with a nice space H, you have your conclusion for $\sigma(K)$.
My question is the follwing is there a general way or sufficient conditions to see if $\sigma(K)=\sigma(P)$. There is a trivial inclusion so the question is better stated as under what conditions does $\sigma(P) \subset \sigma(K)$ holds ?
The motivation behind this question comes from the fact that when I see MCT used in proofs of theorems, it is often the case that the theorem's conclusion is stated for the unbounded real valued functions collection P rather than K (P+the bounded property).