Weil integration formula - Imported Knowlpedia Content

Statement

Let $G$ be a locally compact Hausdorff group, let $H$ be a closed subgroup, and choose left Haar measures $dg$ and $dh$ . There are a positive continuous function $\rho:G\to(0,\infty)$ , satisfying

\rho(gh)=\rho(g)\frac{\Delta_H(h)}{\Delta_G(h)},

and a regular measure $d\dot g$ on the locally compact [[lie-groups/homogeneous-space|homogeneous space]] $G/H$ such that every $f\in C_c(G)$ satisfies

\int_G f(g)\rho(g)\,dg =\int_{G/H}\int_H f(gh)\,dh\,d\dot g.

This is the Weil integration formula for the stated choices and conventions.

Invariant quotient measures

The quotient $G/H$ admits a nonzero $G$ -invariant regular measure exactly when

\Delta_G|_H=\Delta_H.

Under this condition one may take $\rho=1$ , so the formula becomes a literal decomposition of Haar integration into integration along the fibers $gH$ and integration over the quotient. The quotient measure is then unique up to a positive scalar; its normalization changes inversely when the Haar measure on $H$ is rescaled Folland, Chapter 2.

Role of the rho-function

When the modular functions do not agree on $H$ , no invariant quotient measure exists. The rho-function compensates for this mismatch and produces a quasi-invariant measure class on $G/H$ . Different admissible rho-functions give equivalent quotient measures, so constructions based only on the measure class, such as the corresponding unitary quotient representation, do not depend on an accidental choice of density.

Standard cases and conventions

If $H$ is compact, both $\Delta_H$ and $\Delta_G|_H$ are trivial, hence $G/H$ has an invariant measure. Taking $H=\{e\}$ recovers integration on $G$ , while $H=G$ reduces the outer integral to a one-point quotient.

References

G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 2, rho-functions, quotient measures, and Weil's formula.
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Volume I, Springer, 1963. DOI record. Relevant: invariant integration on homogeneous spaces.