Invariant measure on a homogeneous space

Remarks

The idea of the proof: the map $P \colon C_c(G) \to C_c(G/H)$, $(Pf)(xH) = \int_H f(xh) \, dh$, is onto. Under condition 2 the functional $Pf \mapsto \int_G f(x) \, dx$ is well defined on $C_c(G/H)$, and the Riesz representation theorem turns it into the measure $\nu$. Conversely, an invariant $\nu$ gives the functional $f \mapsto \int_{G/H} Pf \, d\nu$, which is a left Haar integral on $G$. Comparing how both sides change under right translation by $h \in H$ forces $\Delta_G(h) = \Delta_H(h)$.

Example: if $H$ is compact, then $\Delta_H$ and $\Delta_G|_H$ are continuous homomorphisms from a compact group into $(0, \infty)$, so both are identically $1$. Thus $G/H$ has an invariant measure; for instance the sphere $S^{n-1} = SO(n)/SO(n-1)$ carries the rotation-invariant surface measure. The same holds when $G$ and $H$ are both unimodular, for instance when $H$ is discrete and $G$ is unimodular.

Non-example: let $G$ be the group of maps $x \mapsto ax + b$ of $\R$ with $a > 0$, and let $H = \set{x \mapsto ax}$. Then $H \cong (0, \infty)$ is abelian, so $\Delta_H = 1$, but $\Delta_G$ is not identically $1$ on $H$. Here $G/H$ is identified with $\R$, and there is no nonzero $G$-invariant Radon measure on $\R$: it would be invariant under translations and dilations at once.

Going deeper

For a student

The theorem reduces the existence of invariant measures to a computation with modular functions, which are often explicit. It applies whenever $H$ is compact, which gives invariant measures on spheres, Grassmannians and Riemannian symmetric spaces. It also applies to lattices $\Gamma$ in unimodular groups, which gives the invariant measure on $G/\Gamma$ used in the theory of automorphic forms. When the condition fails, $G/H$ still has quasi-invariant measures, which change under the action only by a density. The invariant measure is linked to the Haar measures of $G$ and $H$ by Weil's quotient integral formula.