I am trying to proof the following statement. Let $h$ be a bounded, $\mathbb{F}$-predictable process with $\tau$ a $\mathbb{H}$-stopping time, we then like to prove \begin{equation} \mathbb{E}(h_{\tau}\vert\mathcal{F}_{t})=\mathbb{E}\left(\int_{0}^{\infty}h_{u}dF_{u}\Big\vert\mathcal{F}_{t}\right) \ \ \ \ \ \ \ (*) \end{equation} where $F$ is the cumulative distribution function of $\tau$. I have proven this result for $h_{t}=\mathbb{1}_{]v,w]}(t)B_{v}$ where $B_{v}\in\mathcal{F}_{v}$. I would now like to apply the Monotone Class Theorem as defined at:http://planetmath.org/functionalmonotoneclasstheorem. This is where I get stuck. It seems logical to choose $\mathcal{H}$ the set of $\mathbb{F}$-predictable processes $h$ such that $(*)$ holds. But I am rather confused as to how to proceed. (And what should $\mathcal{K}$ be?)
Could anyone please help me?
Ok let's go but I will only outline the solution (using theroem 2 of planetmath.org) :
Take $\mathcal{K}$ as of processes of the form $\sum_{i=1}^n C_i.1[({s_i},{t_i}]\times A_{s_i}]$ where $0\leq {s_i}<{t_i}$, $s_i$ are increasing and $A_{s_i}\in\mathcal{F}_{s_i}$ (those processes are bounded and predictable, and $\mathcal{K}$ is stable by multiplication).
Now take $\mathcal{H}$ as the set of processes that are bounded and predictable and which verify (*). So we have $\mathcal{K}\subset \mathcal{H}$ and $\mathcal{H}$ includes constants.
I'll let you check by use on dominated convergence (or monotone convergence) (which both work well even when conditioning with respect to $\mathcal{F}_t$), that the condition in the bullet point of theorem 2 on $\mathcal{H}$ is satisfied.
This allows you to conclude that $\mathcal{H}$ contains all bounded predictable processes generated by the set of function $\mathcal{K}$, let's call it $\sigma(\mathcal{K})^b$, this in turn contains (prove it) all bounded predictable processes and so we are done because this is what we wanted from the beginning.
Best regards