Premise: Theorem 5.34 from the book "Topics in Optimal Transportation"
Let $X$ be $\mathbb{R}^n$. Let $(T_t)_{0\leq t\leq T_{*}}$ be a locally Lipschitz family of diffeomorphisms in $X$, with $T_{0}=Id,$ and let $v=v(t,x)$ be the velocity field associated with trajectories $(T_{t})$. Let $\mu$ be a probability measure on $X$, and $\rho_t=T_t\#\mu.$ Then, $\rho_t=\rho(t,\cdot)$ is the unique solution of the linear transport equation
$$\frac{\partial \rho}{\partial t}+ \triangledown \cdot (\rho v)=0,\quad 0<t<T_*\ with \ \rho_0=\mu$$
in $C([0,T_*);P(X)),$ where $P(X)$ is equipped with weak topology.
Question: 1) Assuming $d\rho_{t}(x)=f_{t}(x)dx$, How to write equation satisfied by $f_{t}$ from given equation in theorem? 2) How to interpret derivative and divergence operation in equation given that $\rho$ is a measure?