Inducing a Haar measure from a quotient

Suppose that {G} is a locally compact group and that {H} is a closed subgroup with an {H}-left-invariant regular Borel measure {\mu_H} such that {G/H} possesses a {G}-left-invariant regular Borel measure {\mu_{G/H}}. For instance, {G = \mathbf{R}}, {H=\mathbf{Z}}, and {G/H= S^1}. The following is how you then induce a Haar measure on {G}. (Technically, it’s easier to construct Haar measure on compact groups, so this extends that construction slightly.)

For {f\in C_c(G)}, define {T_H(f) : G\rightarrow \mathbf{C}} by

\displaystyle  T_H(f)(x) = \int_H f(xh) d\mu_H.

Let {K=\text{supp}f}. Then {f(xh)}, as a function of {h} is supported on {x^{-1} K}, so the above integral is finite. Moreover, if {x,y\in U} and {U} is compact, then

\displaystyle  |T_H(f)(x) - T_H(f)(y)| = \left| \int_H (f(xh)-f(yh))d\mu_H\right| \leq \mu_H(H\cap U^{-1} K) \sup_h |f(xh)-f(yh)|.

Because a continuous function on a compact set is uniformly continuous (in the sense that there exists a neighbourhood {V} of {e} such that {gh^{-1} \in V} implies {|f(g)-f(h)| <\epsilon}), {T_H(f)} is continous. Since {T_H(f)} is {H}-right-invariant, {T_H(f)} descends to a continuous function {\hat{T}_H(f)} defined on {G/H}. Moreover, if {q} is the quotient map {G\rightarrow G/H}, then {\hat{T}_H(f)} is supported on {q(K)}, so {\hat{T}_H:C_c(G)\rightarrow C_c(G/H)}. Finally, define {\lambda: C_c(G)\rightarrow \mathbf{C}} by

\displaystyle  \lambda(f) = \int_{G/H} \hat{T}_H(f) d\mu_{G/H}.

This linear functional {\lambda} is positive (in the sense that {f\geq0} implies {\lambda(f)\geq0}), so the Riesz representation theorem guarantees the existence of a regular Borel measure {\mu_G} on {G} such that

\displaystyle  \lambda(f) = \int_G f d\mu_G

for all {f\in C_c(G)}. It is now a simple matter to check that {\mu_G} is {G}-left-invariant.

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