A general Weyl-type integration formula for isometric group actions

We show that integration over a G-manifold M can be reduced to integration over a minimal section Σ with respect to an induced weighted measure and integration over a homogeneous space G/N. We relate our formula to integration formulæ for polar actions and calculate some weight functions. In the case of a compact Lie group acting on itself via conjugation, we obtain a classical result of Hermann Weyl.