Liouville's equation and Cauchy problem $\dot{x} = f(x(t),t)$ in optimal control

Question

In a guest lecture about optimal control problem, the speaker introduce the following:

The Liouville's equation encodes a superposition al all classical solutions soving Cauchy problem:

1. Liouville's equation: $$\frac{\partial \mu}{\partial t}+ \mbox{div }(f\mu)=\mu_0-\mu_T$$
2. Cauchy Problem: $$\dot{x}(t) = f(t,x(t))$$

$\mu$ might be of different meanings in different topics; in this lecture, $\mu$ is a occupation measure, $\mu_0$ is the measure evaluated at the initial time. $f(t,x)$ is a nonlinear system. Note: the nonlinear system can actually include the control, i.e., $\dot{x} = f(x,u,t)$ to capture the optimal control problem:

\begin{equation*} \begin{aligned} & {\underset{u}{\text{min}}} & &\int_0^{t_f} l(t,x,u)\\ & \text{s.t.} & & \dot{x}=f(t,x(t),u(t)) \quad \text{a.e. on }[0,t_f]\\ \end{aligned} \end{equation*}

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I am very confused about "Liouville's equation encodes a superposition al all classical solutions of $\dot{x}= f(x,t)$". Could anyone please provide the fundamental background or introduction note about this?

I even have no idea how to relate both stuffs; they look quite different for me.