I understand this statement from the physics side. Consider an $n-$dimensional manifold $\cal M$ ("configuration space") and its cotangent bundle ${\cal P} = T^*\cal M$ ("phase space"), a symplectic manifold. Now consider a scalar density $f:{\cal P}\to\mathbb{R}$ ("phase space distribution function").

An operation which a physicist is interested in is going from the phase space distribution function to the configuration space distribution function. Let's choose coordinates $q^i:{\cal M}\to \mathbb{R}$, $i=1\ldots n$ for ${\cal M}$, and in the usual way, take the canonical coordinates $p_i:{\cal P}\to \mathbb{R}$ so the collection of $q$'s (lifted) and $p$'s are coordinates for ${\cal P}$. Then we define the configuration space distribution function as $$ F(q) \equiv \int d^n p \ \ f(q,p) \,. \qquad\qquad (1) $$ This yields a scalar density of weight 1 on ${\cal M}$.

Above I have done everything in terms of specific coordinate functions, but I feel that it must be possible to state this all geometrically. Over each point $q\in{\cal M}$ there is a fiber $\pi^{-1}(q)$, which is a Lagrangian submanifold ($\cal P$ has a natural Lagrangian fibration).

Question 1: Is there a natural volume form for this fiber? What is the geometric way to define the integral in (1)?

Question 2: Is this possible only for the cotangent bundle of some manifold, or for any symplectic manifold which admits a Lagrangian fibration?

I understand this statement from the physics side. Consider an $n-$dimensional manifold $\cal M$ ("configuration space") and its cotangent bundle ${\cal P} = T^*\cal M$ ("phase space"), a symplectic manifold. Now consider a scalar density $f:{\cal P}\to\mathbb{R}$ ("phase space distribution function").

An operation which a physicist is interested in is going from the phase space distribution function to the configuration space distribution function. Let's choose coordinates $q^i:{\cal M}\to \mathbb{R}$, $i=1\ldots n$ for ${\cal M}$, and in the usual way, take the canonical coordinates $p_i:{\cal P}\to \mathbb{R}$ so the collection of $q$'s (lifted) and $p$'s are coordinates for ${\cal P}$. Then we define the configuration space distribution function as $$ F(q) \equiv \int_{\pi^{-1}(q)} d^n p \ \ f(q,p) \,. \qquad\qquad (1) $$ This yields a scalar density of weight 1 on ${\cal M}$.

Above I have done everything in terms of specific coordinate functions, but I feel that it must be possible to state this all geometrically. Over each point $q\in{\cal M}$ there is a fiber $\pi^{-1}(q)$, which is a Lagrangian submanifold ($\cal P$ has a natural Lagrangian fibration).

Question 1: Is there a natural volume form for this fiber? What is the geometric way to define the integral in (1)?

Question 2: Is this possible only for the cotangent bundle of some manifold, or for any symplectic manifold which admits a Lagrangian fibration?

EDIT: As best as I can tell, $d^np$ is best defined as $$ d^np \equiv \iota_{e_1}\iota_{e_2}\cdots\iota_{e_n}\mathrm{dVol} $$ where $e_i\equiv \partial/\partial q^i$, and where $$ \mathrm{dVol} \equiv \omega^{\wedge n}\\ = dq^1 \wedge \cdots \wedge dq^n \wedge dp^1 \wedge \cdots \wedge dp^n $$ the symplectic 2-form wedged with itself $n$ times to make the top-rank volume form. It's still not clear to me how to define this all in a geometric way without regard to coordinates, or at least to show that it is coordinate independent; and if this can be done on any symplectic manifold with Lagrangian fibration.