= Finite-dimensional Faddeev-Popov gauge reduction
{title2=$\int dx\,f(x)=\operatorname{vol}(G)\int dx\,\delta(F(x))|\det M(x)|f(x)$}
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 $\int_Gd\mu(g)\,\delta(F(x_g))=1/|\det M(x_0)|$. 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.
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