Markov process characterization and application of monotone class theorem

Question

I'm studying from Continuous Martingales and Brownian Motion, Daniel Revuz & Marc Yor the following proposition about the characterization of Markov process:

Where

1. $X_t \colon (\Omega, \mathcal{F},P) \to (E,\mathcal{E})$ for all $t$ in the time set;
2. $\mathcal{E}_+$ is the set of the positive measurable function over the state space $(E,\mathcal{E})$;
3. $(\mathcal{F}_t^0)$ is the natural filtration generated by the process.

To prove the sufficient condition, the monotone class theorem is used:

I guess I have to prove that the class $$ \left\{ \prod_{i=1}^k f_{k}(X_{t_k}) \; \colon \; t_1 < \ldots < t_k \leq t, \; f_i \in \mathcal{E}_+ \right\} $$ is closed under intersection and generates the $\sigma$-algebra $\mathcal{F}^0_t$, but I can't figure out how to do it.