X_n]} \\
&= P[y_n | X_n],
\end{split}
\label{prob1}$$ which is, in fact, independent of $Y_{n-1}$.

We then discuss the mutual information obtained by multiple measurements on nonequilibrium dynamics. Suppose that we obtain measurement outcomes $Y_{n-1}$ at time $t_{n-1}$. If we perform another measurement at time $t_n$ and obtain outcome $y_n$, we obtain the mutual information between $y_n$ and $X_n$ under the condition that we have obtained $Y_{n-1}$: $$\begin{split}
I[y_n : X_n | Y_{n-1}] &:= \ln \frac{P[y_n | X_n, Y_{n-1}]}{P[y_n | Y_{n-1}]} \\
&= \ln \frac{P[y_n | X_n]}{P[y_n | Y_{n-1}]},
\end{split}$$ where we used Eq. (\[prob1\]). We note that, if the measurement is Markovian, $I[y_n : X_n | Y_{n-1}]$ reduces to $I[y_n : x_n | Y_{n-1}]$. We denote as $I_{\rm c}$ the sum of these mutual information contents obtained by multi-measurements, that is, $$\begin{split}
I_{\rm c} [ X_n: Y_n] &:= \sum_{k=1}^{M'} I [y_{n_k}: X_{n_k}] \\
&= \ln \frac{P[Y_n | X_n]}{P[Y_n]},
\end{split}
\label{mutual1}$$ where we used Eq. (\[prob0\]). We note that the same quantity has been discussed in Ref. [@Cao2]. From Eq. (\[mutual1\]), we find that $I_{\rm c} [Y_n : X_n]$ equals the mutual information between trajectories $X_n$ and $Y_n$ defined as $I [Y_n : X_n ] := \ln (P[Y_n | X_n] 