 Finite-dimensional Faddeev-Popov gauge reduction

Assume an invariant volume measure, an invariant integrand and one regular gauge slice representative per orbit. At that representative, let $M$ be the Jacobian matrix of $F$ along the group coordinates, whose Haar measure is normalized to ordinary coordinate measure at the identity. Changing coordinates from the group parameter to $F$ gives ${\int }_{G}d\mu (g)\delta (F(x_g))=1/|\det M(x_0)|$. Multiply by its reciprocal and insert this identity into the original integral. Invariance permits a change of variables $x\mapsto x_g$ for each $g$, factoring out the group volume. The absolute determinant is required for ordinary unoriented real integration; a consistently positive orientation permits writing $\det M$. Infinite group volume is divided out formally or treated with a regulator.