Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if $J:M\to \frak g^*$ be its equivariant moment map then the reduced space is $$S=J^{-1}(\mu)/G_\mu$$ where $\mu\in \frak g^*$ and $G_\mu$ is isotropy group at point $\mu$ .

What is the relation between $Vol(S)$ and $Vol(M)$