I have a lot of trouble in applying the functional monotone class theorem. Therefore I'm solving some exercises to get some experience. Maybe someone could help me with the following. Suppose I have shown that the following equality is true:
$E[g f(W_{t+h}-W_t)]=E[g]E[f(W_{t+h}-W_t)]$
for a bounded and measurable function $g$ and for a bounded and continuous function $f$ on $\mathbb{R}$. Now I should use the functional monotone convergence theorem to extend this to all f, which are bounded and measurable on $\mathbb{R}$. According to PlanetMath (Theorem 2), how would you choose $\mathcal{K}$ and $\mathcal{H}$ in this situation. Since I have trouble to apply the theorem, it would be appreciated if someone could help me to see a example of its use.
math