the trajectory of the system before $t_n$ due to the causality. Here we assumed that the property of the measurement error at time $t_n$ does not explicitly depend on $Y_{n-1}$ or $P[X_n]$. This assumption is also justified in many real experimental situations.

2.  The unconditional probability distribution of $X_n$, $P[X_n]$, is not affected by the back-action of the measurement. Since the system is classical, this assumption is justified for many real systems such as colloidal particles and macromolecules.

If $P[y_n | X_n] = P[y_n | x_n]$, we call the measurement Markovian, which means that the outcome is determined only by the system’s state immediately before the measurement. This condition is satisfied if the measurements can be performed in a time interval that is sufficiently shorter than the shortest time scale $\Delta t$ of the system. We note that the Markovness of the measurement is independent of that of the dynamics.

We assume that the measurements are performed at times $t_{n_1}$, $t_{n_2}$, $\cdots$, $t_{n_M}$, where $0 \leq n_1 < n_2 < \cdots < n_M \leq N$. If $n_1 = 0$, $n_2 = 1$, $n_3 = 2$, $\cdots$, $n_{N+1} = N$ hold, the measurement is time-continuous in the limit of $\Delta t 