Hello,

I want to show by using a monotone class argument that the following equality (i.e. it is a martingale) holds for all bounded and \mathcal{G}_s-measurable random variables Y_s:
E[Y_s(W_t-W_s)]=0, s\leq t. (*)
where W is an \mathcal{F}_s Brownian motion (and it follows from (*) also a \mathcal{G}_s Brownian motion).

\mathcal{G}_s is the enlargement of the filtration \mathcal{F}_s by the filtration generated by a r.v. X. So far I have shown that
(*) holds for Y_s=f(X)H_s where f is bounded and measurable; and H_s is bounded and \mathcal{F}_s-measurable. Now, I think I need a monotone class argument in order to show that (*) holds for all bounded and \mathcal{G}_s-measurbale random variables.

But unfortunately I am not very familiar in using monotone class arguments.
How do I find the sets needed in the monotone class theorem?

Thanks in advance.