Ok, I'm currently confused because of two different definitions for the Liouville measure associated to a smooth manifold $M$ of dimension $n$. These are:
a) The measure $\mu$ on the cotangent bundle $T^*M$ induced by the volume form $(d\alpha)^n$ where $\alpha$ is the canonical 1-form on $T^*M$ and $\omega:=d\alpha$ makes $T^*M$ a symplectic manifold.
This $\mu$ is invariant under any Hamiltonian flow (that is, any flow that integrates a vector field $X_H$ defined for some smooth $H:T^*M\to \mathbb{R}$ by $\theta(X_H)=\omega(\theta,dH)$ for all $\theta\in T^*(T^*M)$ ).
b) The measure $\nu$ that may be defined, once $M$ is endowed with a Riemannian metric $g$, on $SM$, the unit tangent bundle. This measure is given locally by the product of the Riemannian volume on $M$ (i.e. $\text{det}(g_{ij})dx_1\wedge\dots\wedge dx_n$) and the usual Lebesgue measure on the unit sphere.
This $\nu$ is invariant under the geodesic flow on $SM$.
Now, what is, if any, the relationship between these measures?
I imagine that once $M$ is endowed with $g$ we can identify $T^*M$ with $TM$ ($q,p\to q,v$ iff $p(v')=g(v,v')$ for al $v'\in T_qM$). Will then the restriction of $\mu$ to the unit cotangent bundle (assuming this restriction to a submanifold of smaller dimension is well defined) correspond to $\nu$? It's the only reason I could imagine for giving the same name to two different measures defined on two different spaces ($T^*M$ and $SM$, respectively), but I could not find such a statement on any reference book.
Best Answer
I can't say for sure that it is the reason why the two notions are named the same, since they are not equivalent, but you have identified a relation between the two. In a way, the second notion is a special case of the first.
Let $(M,g)$ be a Riemannian manifold. As such, there exists a monomorphism of vector bundles
$$\flat : TM \to T^*M : v \mapsto v^{\flat} = v \, \lrcorner \, g \; .$$
Since both bundles have the same (finite) rank, it is also an epimorphism and so an isomorphism : there exists an inverse morphism
$$\sharp : TM \to T^*M : \lambda \mapsto \lambda^{\sharp} \; .$$
We can define a bundle metric $g^*$ on $T^*M$ as follows : for $\alpha, \beta \in T^*_mM$, let $g^*(\alpha, \beta) = g(\alpha^{\sharp}, \beta^{\sharp})$. This means that $\flat$ and $\sharp$ are "bundle isometries" : $(TM, g)$ is somewhat the same as $(T^*M, g^*)$. As such, the map $\flat$ sends the Liouville measure of and the geodesic flow on $(TM, g)$ to what we might consider the Liouville measure of and the geodesic flow on $(T^*M, g^*)$.
From a different point of view, $g^*$ defines a smooth "bundle quadratic form" $Q^*$ on $T^*M$ : given $\alpha \in T^*_mM$, let $Q^*(\alpha) = \frac{1}{2} \, g^*(\alpha, \alpha)$. This is a smooth real function of $T^*M$, in fact a "Hamiltonian" function if we equip $T^*M$ with its canonical symplectic form. There is an associated Hamiltonian vector field $X_{Q^*}$ whose flow preserves the level sets of $Q^*$, that is, the sphere-subbundles of $T^*M$ of different $g^*$-radii. Furthermore, this flow happens to be the same as the above "geodesic flow" on $(T^*M, g^*)$.
Since this flow preserves both the canonical volume form on $(T^*M, \omega_0)$ and the Liouville measure on $(T^*M, g^*)$, both volume forms have to be proportional to one another by a (nonvanishing) function $f$ which is constant along the flow. A priori, there is no reason for $f$ to equal 1 : after all, the canonical volume form knows nothing about the metric $g$ on $M$. However, in order to restrict the canonical volume form to a volume form on the level sets of $Q^*$, one needs to take the interior product with (for instance) a normal vector field, which necessitates a Riemannian metric on $T^*M$. I don't know if it is true, but may be there is a way to get the same measure on the level sets from both approaches.
If you take a look at the Wikipedia article on Liouville's theorem, you can read the following :
This suggests that the names "Liouville measure" might be only honorific too, since proofs of "the geodesic flow preserves the measure $\nu$" and "the Hamiltonian flow preserves the measure $\mu$" exist which use an identity of Liouville.