Monotone class theorem I have some question about the Monotone Class Theorem and its application. First I state the Theorem:

Let $\mathcal{M}:=\{f_\alpha; \alpha \in J\}$ be a set of bounded functions, such that $f_\alpha:N\to \mathbb{R}$ where $N$ is a set. Further we suppose that $\mathcal{M}$ is closed under multiplication and define $\mathcal{C}:=\sigma(\mathcal{M})$. Let $\mathcal{H}$ be a real vector space of bounded real-valued functions on $N$ and assume:
  
  
*
  
*$\mathcal{H}$ contains $\mathcal{M}$
  
*$\mathcal{H}$ contains the constant function $1$.
  
*If $0\le f_{\alpha_1}\le f_{\alpha_2}\le \dots$ is a sequence in $\mathcal{H}$ and $f=\lim_n f_{\alpha_n}$ is bounded, then $f\in   
    \mathcal{H}$
  
  
  Then $\mathcal{H}$ contains all bounded $\mathcal{C}$ -measurable functions.

My first question: I know the Dynkin lemma which deals with $\sigma$-Algebras and $\pi$-Systems. Which is the stronger one, i.e. does Monotone Class imply Dynkin or the other way around?(or are they equal?) A reference for a proof would also be great!
My second question is about an application. Let $X=(X_t)$ be a right continuous stochastic process with $X_0=0$ a.s. and denote by $F=(F_t)$ a filtration, where $F_t:=\sigma(X_s;s\le t)$. I want to show:

If for all $0\le t_1<\dots<t_n<\infty$ the increments $X_{t_{i}}-X_{t_{i-1}}$ are independent then $X_t-X_s$ is independent of $F_s$ for $t>s$.

The hint in the book is to use Monotone Class Theorem. So
$$\mathcal{H}:=\{Y:\Omega\to \mathbb{R} \mbox{ bounded };E[h(X_t-X_s)Y]=E[h(X_t-X_s)]E[Y] \forall h:\mathbb{R}\to\mathbb{R} \mbox{ bounded and Borel-measurable}\}$$
This choice is clear. Now they say 
$$\mathcal{M}:=\{\prod_{i=1}^n f_i(X_{s_i});0\le s_1\le \dots\le s_n\le s,n\in \mathbb{N},f_i\colon\mathbb{R}\to\mathbb{R} \mbox{ bounded and Borel-measurable}\}$$
Two question about this choice:


*

*Why is $\sigma(\mathcal{M})=\mathcal{F}_s$?

*Why do they define $\mathcal{M}$ like this? (As family of products?)


Thanks in advance!
hulik
 A: 1)
Monotone class theorem and $\pi-\lambda$-system of Dynkin are complementary ways to prove that a certain set of subsets contains a $\sigma$- algebra.
One can show that $M(G)$ the smallest monotone class of an algebra $G$ is a $\lambda$- system. Similarly, one can show that $\lambda(P)$ the smallest $\lambda$- system of a $\pi$- system $G$ is a monotone class.
The point is to see which is the simpler criterion.
Q:It is easier to check that $G$ is an algebra or to check that it is a $\pi$- system?A: It is easier to check that $G$ is a $\pi$ system (every algebra is a $\pi$-system the converse does not follow)
Q:It is easier to check that $M$ is a monotone class or to check that it is a $\lambda$- system?A: It is easier to check that $M$ is a $\lambda$ system (every algebra is a $\lambda$-system the converse does not follow)
2) 
 To see that $\sigma(M) = \mathcal{F}_s$ note that you can approximate $1_{A_{s_1}}(X_1) 1_{A_{s_2}}(X_2) \ldots 1_{A_{s_k}}(X_k)$ with continuous functions $f_1(X_{s_1})f_2(X_{s_2})\ldots f_k(X_{s_k})$ (see  https://en.wikipedia.org/wiki/Urysohn%27s_lemma)
the reason you choose the family of product of continuous functions is that one often deals better with properties for continuous functions.(in fact one may only need a denumerable set of continuous functions , depending on the problem at hand this is very useful)
note that one can know a measure by it's values on measurable sets ($\{\mu(A), A \in \mathcal{F}\}$) But one can also know a measure by it's values on continuous functions $\{\mu(f) = \int f \, d\mu, f \in C(X)\}$ when $X$ is a  locally compact Hausdorff space. (see https://en.wikipedia.org/wiki/Riesz%E2%80%93Markov%E2%80%93Kakutani_representation_theorem)
