$\textbf{Ordinary Differential Equation:}$ Let $x(\cdot) \in C([0,1];\mathbb{R}^n)$ (C($\cdot$) denotes the set of continuous functions) be the trajectories satisfying the following differential equation $$\begin{gather}\label{eq:ode} \dot{x}(t) = f(t,x(t))\quad x(0) = x_0\in \mathbb{R}^n \end{gather} $$ such that $ (t,x)\in X ~, ~\dot{x}(t) \in V ~\forall t\in[0,1]$. $f(t,x)$ is called the vector field and we here assume that it is Lipschitz regular.
$\textbf{Measures and Continuity equation:}$ Now consider the linear functionals $\mu$ on $C(X \times V)$ such that $\mu \in C^*(X \times V)$ and $\gamma \in C^*(X)$, \begin{gather}\label{eq:cont_eqn} \langle \mu, \partial_t \phi(t,x) + \partial_x\phi(t,x)\cdot v \rangle = \int \phi d\gamma -\phi(t_0,x_0) \end{gather} for all $\phi\in C^1(\mathbb{R}^{n+1}) $.
Note: This is known as the continuity equation and the well-posedness of the associated Cauchy problem are related to that of differential equation mentioned above.
$\textbf{Extreme points of set of Measures}$
Consider a set of such linear functional which is bounded, i.e., \begin{gather} W := \{(\mu, \gamma)| ~{\rm s.t.~}\quad \mu,\gamma \ge 0, \quad ||\mu||\le 1,||\gamma|| \le 1 \quad and ~{\rm satisfies ~} {\rm continuity~ equation} \} \end{gather}
It is easy to prove that set $W$ is convex and weak* compact.
$\textbf{What will be the extreme points of set} W?$ I speculate that all possible extreme points are dirac measures supported on the admissible trajectories to the differential equations. But I am not able to prove the "all" part of the statement. I will be grateful for any ideas or pointers.