What does the functional monotone class theorem say and how does it relate to the other monotone class theorem?

This is a question in 2 steps :

1. What does the monotone class theorem give us ?
   I have been thinking about the functional version of the theorem for a while now and I can not see really what it gives us. Let’s write it down :

Let $H$ be a vector field of bounded functions $Omega rightarrow mathbb{R}$. Assume that $H$ contains the constant functions and has the following property : If $(f_n)$ is an increasing and bounded function series of positive functions of $H$ then $f:=limf_n in H$. Let $C$ be a subset of $H$ stable under multiplication. Then $H$ contains all bounded $sigma(C)$-measurable functions i.e. $L_b^0(sigma(C)) subset H$

Now, if we take a bounded $sigma(C)$-measurable function, it is also a function in $H$. But the properties of $H$ are too vague and not very useful for us, right ? It does not have any property related to measurability.

I know that this theorem is used to prove that if two probability functions are equal on a generating part, stable under $cap$ then they are equal on the whole set as well. But how can we see that on this theorem ?

2. My second question is : how does it relate to the other monotone class theorem ?

If $A$ is an algebra on a set $Omega$, the the smallest monotone class containing $A$ is equal to the smallest sigma-algebra containing $A$

I can see that the property on $H$ saying that if $(f_n)$ is an increasing and bounded function series of positive functions of $H$ then $f:=limf_n in H$ is written monotone class all over it, but do we have that the smallest monotone class containing $C$ is equal to the smallest sigma-algebra containing $C$ ? Is that the point of the functional monotone class theorem ?