I am trying to understand the proof given here (28.4) but am having some significant problems. It is clear that it suffices to show $$\int_0^t H_sG_s d \langle M, N \rangle_s \leq \sqrt{\int_0^t H_s^2 d \langle M \rangle_s\int_0^t G_s^2 d \langle N \rangle_s} \quad \quad \textbf{(1)}$$
for every bounded $\mathscr{B}(0 \infty) \otimes\mathcal{F}$ measurable functions $G, H$. It is very clear from the link that (1) holds for all functions of the form $$G_s = \sum_{i=1}^n g_i \textbf{1}(s \in (t_{i-1}, t_i]), H_s = \sum_{i=1}^n h_i \textbf{1}(s \in (t_{i-1}, t_i])$$ for $g_i, h_i$ bounded $\mathcal{F}$ measurable.
Then it just states that "by monotone class arguments" this (1) holds for all bounded measurable functions, but I do not see how this is the case. Fixing $H$ in the above form, the set of G for which (1) holds clearly contains the set of indicators of a $\pi$ system which generates $\mathscr{B}(0 \infty) \otimes\mathcal{F}$ but I do not believe it is a vector space (if it were we could indeed apply the monotone class theorem as the book states). Does anyone know how to reconcile this?
I'm not sure if the following idea is correct: We should note that (1) holds for all bounded continuous processes by the dominated convergence theorem (as we may approximate such $G,H$ by taking $g_i = G_{t_i}, h_i = H_{t_i}$ for a partition of $[0,t]$ with norm tending to 0). For fixed $\omega$, the set of continuous functions is dense in the set of bounded functions and so we may approximate any bounded function with a continuous one, and this suffices. Is this correct? If so, why does every resource I look at say to use the monotone class theorem when it's not applicable?