2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/216339We 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 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.14 pages, based on the authors docotral thesisDifferential Geometry57S25; 53C20text