Martingale representation theorem for reference:
Theorem: (Martingale Representation) Let $M$ be a square integrable Brownian martingale with $M_0 = 0$.Then there exists a process $X$ which is progressively measurable (with an extra constraint not needed here) such that $M = X \cdot W$. Where $W$ is a Brownian motion to which we integrate.
Where the "dot" stands for stochastic integral notation. In many proofs one can prove the theorem by starting from an orthogonal decomposition $M=N+Z$. And then prove that for any Borel-measurable functions $f_k$ and $n \in \mathbb{N}$ with $k=1,2,\ldots,n$ and time points $0\leq s_1 < s_1 < \ldots < s_n \leq t$ we have that $$\mathbb{E} \left[ Z_t \prod_{k=1}^n f_k(W_{s_k}) \right]=0.$$
Then after proving this many proofs say that by a monotone class argument we have that $\mathbb{E} \left[ Z_t \xi \right]=0$ for all $\xi$ that are bounded $\mathcal{F}^W_t$-measurable functions. But I don't have any clue to how this argument should work exactly. I know that by Monotone class theorem, that if we have a $\pi$-system $\mathcal{A}$ and a function space $\mathcal{H}$ which is a vector space (indicators, sum) and if a sequence of non-negative functions increasing to a bounded function then also is this function contained in the space then it contains all bounded functions that are measurable with respect to $\sigma(\mathcal{A})$, the $\sigma$-algebra generated by $\mathcal{A}$.
Could anyone elaborate this monotone class argument wrt $\xi$? My thanks for any assistance.
For $k \in \{1,\ldots,n\}$ let $U_k \subseteq \mathbb{R}$ be an open relatively compact set. By Urysohn's lemma, there exist sequences $(f_j^k)_{j \in \mathbb{N}} \subseteq C_c(\mathbb{R})$, $k \in \{1,\ldots,n\}$, such that
$$0 \leq f_j^k \uparrow 1_{U_k}.$$
Using the monotone convergence theorem, we get
$$\mathbb{E} \left( Z_t \prod_{k=1}^n 1_{U_k}(W_{s_k}) \right)=0. \tag{1}$$
By considering $\tilde{U}_k := U_k \cap B(0,R)$ for fixed $R>0$, we obtain $(1)$ also for open sets $U_k$ (not necessarily relatively compact). The family
$$\mathcal{D} := \{A \in \mathcal{F}_t^W; \mathbb{E}(Z_t 1_A)=0\}$$
defines a Dynkin system and contains sets of the form
$$\bigcap_{j=1}^n \{W_{s_j} \in U_j\}$$
for arbitrary $n \in \mathbb{N}$, $0 \leq s_1 < \ldots < s_n \leq t$ and open sets $U_j$. Since these sets are a $\cap$-stable generator of $\mathcal{F}_t^W$, we get $\mathcal{D} := \mathcal{F}_t^W$. Finally, we can approximate any bounded $\mathcal{F}_t^W$-measurable function by ($\mathcal{F}_t^W$-measurable) simple functions and the claim follows from the dominated convergence theorem.