@egreg: I just want to put this text
Let $(E,\e)$ be an arbitrary measurable space, which is meant to be the state space of a dtMP. For each $n\in \N_0$ the space of trajectories of the process of the horizon $n$ we define as $(\Omega_n,\f_n)$ where $\Omega_n = E^{n+1}$ and $\f_n = \e^{\otimes (n+1)}$ is its product $\sigma$-algebra.
Into a desired font.
Here \e is $\mathcal E$ and \f is $\mathcal F$