I read the proof of Theorem 4.2 in Markov Processes in Ethier and Kurtz (Chapter 4, pages 184-186) , and I have a question in one detail in the proof that I cannot explain. The theorem essentially states that if any two solutions to a martingale problem have the same one-dimensional distributions, then they have the same finite dimensional distributions.
In the part of the proof where uniqueness is proved, it is said that it is sufficient to consider positive functions. Why is that? Are the positive bounded functions on a metric space separating? and why?
Write $f_k=f_k^{(0)}-f_k^{(1)}$, where $f_k^{(0)}(x)=\max\{f_k(x),0\}$ and $f_k^{(1)}(x)=\max\{0,-f_k(x)\}$. Suppose that we proved (4.19) in the case where the functions $f_k$ are non-negative. Then \begin{align} \mathbb E^{P}\left[\prod_{k=1}^m f_k(X(t_k))\right]&=\mathbb E^{P}\left[\prod_{k=1}^m \left(f_k^{(0)}(X(t_k))-f_k^{(1)}(X(t_k))\right)\right]\\ &=\sum_{\delta_1,\dots,\delta_m\in \{0,1\}}(-1)^{\sum_{\ell=1}^m\delta_\ell}\mathbb E^{P}\left[\prod_{k=1}^mf_k^{(\delta_k)}(X(t_k)) \right]\\ &=\sum_{\delta_1,\dots,\delta_m\in \{0,1\}}(-1)^{\sum_{\ell=1}^m\delta_\ell}\mathbb E^{Q}\left[\prod_{k=1}^mf_k^{(\delta_k)}(X(t_k)) \right] \mbox{ by (4.19) for non-negative functions}\\ &=\mathbb E^{Q}\left[\prod_{k=1}^m f_k(X(t_k))\right]. \end{align}