Given a (Hamiltonian) flow $\phi^t:\mathbb{R} \times D \rightarrow D$ on some compact set $D\subset \mathbb{R}^n$ ($D$ can be seen as some level set of the Hamiltonian), and let $\{U_\alpha\}$ be a finite cover of $D$.
Suppose on each $U_\alpha$, there exists a measure $\mu_\alpha$ which is invariant under the flow, i.e. given $A\subset U_\alpha$ and $T$ that are small enough, we have $ |t|\leq T,\; \phi^t(A) \subset U_\alpha$ and $$\quad \mu_\alpha(A) = \mu_\alpha(\phi^t(A)).$$
Is there a way to define a global invariant measure on $D$? If not, what additional information I will need to do this?