Density of cylindrical random variables in classical Wiener space I'm currently working on Malliavin calculus, and a theorem in my class notes is bothering me :
Denote W the Wiener space of continuous functions from $[0,1]$ to $\mathbb{R}$, and $\mu$ the associated Wiener measure. Let also the coordinate random variables (Brownian motion) such that $W_t(w)=w_t$. The theorem is the following :
Theorem : The random variables $\{f(W_{t_1},\dots,W_{t_n}),t_i\in [0,1], n\in\mathbb{N},f\in\mathcal{S}(\mathbb{R}^n)\}$ are dense in $L^2(\mu)$. ($\mathcal{S}(\mathbb{R}^n)$ is the Schwarz space.
The proofs says : it follows from Martingale convergence theorem and monotone class theorem. But I can't figure how the use these theorems to get the result... In addition, I think there is kind of a scheme to prove density of random variables with monotone class theorem, so if someone could explain the general idea, it would be very helpful.

I don't know how to use the hint.  But you could use the fact that the underlying sigma-algebra is generated by sets of the form $\{w : w_t \in [a,b]\}$ for all $t\in[0,1]$ and $a,b \in \mathbb R$.
