  We show that integration over a $G$-manifold $M$ can be reduced to
integration over a minimal section $\Sigma$ with respect to an induced weighted
measure and integration over a homogeneous space $G/N$. We relate our formula
to integration formulae for polar actions and calculate some weight functions.
In case of a compact Lie group acting on itself via conjugation, we obtain a
classical result of Hermann Weyl. Our formula allows to view almost arbitrary
isometric group actions as generalized random matrix ensembles. We also
establish a reductive decomposition of Killing fields with respect to a minimal
section.
