Please see below the section from the book Stochastic Integration and Differential Equations by P. Protter containing Theorem 43 of Chapter 4. I'd assume you'd need to be familiar with the book to answer the following questions I have. Please correct me where I am having wrong assumptions.
1) He does not develop his notion of predictable representability for finite time-horizons. How can he claim (in the red box) the set $\mathcal{A}$ of Brownian motion's stopped at $t_0$ to have the representation property? He avoids an obvious later conflict by assuming $M=M^{t_0}$ in the green box. In the same way I don't get how he establishes/means $P=Q$ by looking only at $\mathcal{F}_{t_0}$. How does "assuming $X$ to be stopped at $t_0$" make the claim $\mathcal{M}^2(\mathcal{A})=\{P\}$ rigorous? What does he leave to the reader at that point?
2) Still in the red box: I am not familiar with the argument using the bounded Borel functions to show that $P=Q$, how does it work? Apart from my issues from question 1), is it standard and/or mentioned somewhere in the rest of the book?
3) In the proof of Corollary 1 in the orange box, how can he claim $P$ to be extremal for $\mathcal{A}=\{X_1,\dots,X_n\}$ where $X_i$ are Brownian motions? They are not $L_2$-martingales, and the book didn't develop the notion of extremal points for sets stemming from martingales not contained in $M^2$, the space of $L_2$ martingales. And either way, he should refer to $\mathcal{M}^2(\mathcal{A})$ (not $\mathcal{A}$) when speaking of $P$ as an extremal point. I don't doubt that the notion can be extended to local $L_2$ martingales, does he assume the reader to think of that? If yes, I figure the statement is valid because it was shown in the proof of theorem 43 that $P|_{\mathcal{F}_t} = Q|_{\mathcal{F}_t}$ for all $t$ (and $\text{domain}(P)=:\mathcal{F}\supset\mathcal{F}_\infty$ is no problem because?)
Thank you very much! I don't doubt that most of my issues can be overlooked, I just want to make things rigorous (at my level).
