Let M be a symplectic manifold with symplectic 2-form $\Omega$ provided with an action of a compact connected Lie group G with Lie algebra $\mathfrak{g}$. Suppose that this action is Hamiltonian. Let $\mu :M \rightarrow \mathfrak{g}^*$ be the moment map associated to this action, and denote by $dm_L:= \frac{\Omega^n}{n!}$, $dim M =2n$, the Liouville volume form on M.
My questions are about the pushforward $\mu_*(dm_L)$: it appears a lot in the texts which speak about symplectic geometry (that I'm not very familiar with!). My first question Is What is its definition, I know What is the pullback of a a differential form but never knew about what is the pushforward of a differential form?
My second question is what is the intuition behind this definition of $\mu_*(dm_L)$, and why it is important in symplectic geometry?
Thank you!