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
Best Answer
I think $\mathcal K = C_b(\mathbb R)$ and $$\mathcal H = \{f \text{ bounded measurable}\, ;\; \text{($\ast$) holds for all bounded measurable } g \}$$ where \begin{equation}\tag{$\ast$} E[gf(W_{t+h} - W_t)] = E[g]E[f(W_{t+h} - W_t)] \end{equation} would work, using monotone (or dominated) convergence to show closedness of $\mathcal H$ under monotone limits.
Note that $\mathcal K$ generates the Borel-$\sigma$-algebra, because any closed set $C$ is the zero set of a bounded continuous function, e.g. of $x \mapsto \min\{1,d(x,C)\}$, where $d(x,C) = \inf\{d(x,y)\mid y \in C\}$.