measurement [@Shannon; @Cover-Thomas]. The Shannon information contents of the probability variables are given by $$H_x := -\sum_x P[x] \ln P[x], \ H_y := - \sum_y P[y] \ln P[y],$$ which characterize the randomnesses of $x$ and $y$, respectively. On the other hand, the mutual information content $\langle I \rangle$ between $x$ and $y$ is given by $$\langle I \rangle := \sum_{xy} P[x,y]I[x:y],$$ where $$I[x:y] := \ln \frac{P[y|x]}{P[y]}.$$ In this paper, we also call $I[x:y]$ the mutual information. We note that $I[x:y] = I[y:x]$ holds due to the Bayes theorem (\[Bayes\]).

The mutual information $\langle I \rangle$ measures the amount of information obtained by the measurement. It is known that $$0 \leq \langle I \rangle \leq H_x, \ 0 \leq \langle I \rangle \leq H_y.$$ If the measurement is error-free, $\langle I \rangle = H_x = H_y$ holds.

Measurements on Nonequilibrium Dynamics
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We next formulate multiple measurements on nonequilibrium dynamics, and discuss the properties of the mutual information obtained by the measurements.

Let $y_n$ be the outcome at time $t_n := n \Delta t$. In this section, we assume the followings:

1.  The error of the measurement at time $t_n$ is characterizes by $P[y_n | X_n]$, where $y_n$ can depend on 