We know $\mathcal{F}(t)=\sigma(\{W(s):0\leq s \leq t\})$ is the smallest $\sigma$ algebra for which the Brownian Motion $W(s)$ is measurable. We are given these definitions:
$$ X(w) = \lim_{n\rightarrow \infty} X_n(\omega) \text{ where } X_n=g_n(W(t_0^n),W(t_1^n),...,W(t_{N_n}^n))$$ $$ \text{ and } Y=W(t+\delta)-W(t) $$
Let $\phi(x) \text{ and } \psi(x) $ denote any bounded, continuous functions. Use $$f(W(t_0),W(t_1)-W(t_0),...,W(t_{N})-W(t_{N-1}))=g(W(t_0),W(t_1),...,W(t_{N})) $$ and independence of the increments to show $$ E(\phi(X_n)\psi(Y))=E(\phi(X_n))E(\psi(Y)).$$
I understand how independence and expected value come into play, but I'm confused as to how $\phi$ and $\psi$ play a role in this.