Weil's quotient integral formula

Remarks

The inner integral depends only on the coset $xH$, by left invariance of $dh$, so the right side makes sense. The proof is the construction in the theorem on the invariant measure on a homogeneous space. The formula is taken as the definition of a functional on $C_c(G/H)$, and the hypothesis on the modular functions shows that it is well defined. Uniqueness holds because every function in $C_c(G/H)$ has the form $xH \mapsto \int_H f(xh) \, dh$ for some $f \in C_c(G)$. The formula then extends to all $f \in L^1(G)$, with the inner integral defined for almost every coset.

Example: $G = \R$, $H = \Z$, with Lebesgue measure on $\R$ and counting measure on $\Z$. Then $G/H$ is the circle $\R/\Z$, the invariant measure is Lebesgue measure on $[0, 1)$, and the formula reads $$\int_\R f(x) \, dx = \int_0^1 \sum_{n \in \Z} f(x + n) \, dx.$$ This is the identity behind periodization and the Poisson summation formula.

Without the hypothesis on the modular functions, $G/H$ has no nonzero $G$-invariant Radon measure. The formula then holds only in a modified form, with a quasi-invariant measure and a correcting density.

Going deeper

For a student

The formula is a Fubini theorem for the fibration $G \to G/H$ with fibres the cosets $xH$. It is used to normalize measures on homogeneous spaces compatibly with Haar measures, for instance to compute the volume of $SL_n(\R)/SL_n(\Z)$. For a lattice $\Gamma$ in $G$, it relates integrals over $G$ to integrals over a fundamental domain, which is basic in the theory of automorphic forms. For abelian $G$, combined with Pontryagin duality, it gives the Poisson summation formula for a closed subgroup and its annihilator.