# Haar measure of quotient group

Suppose $G$ is a (Hausdorff) compact group with normalised Haar measure $\mu$, and that $H\trianglelefteq G$ is a closed normal subgroup. Is it true that the pushforward of $\mu$ to $G/H$ is the normalised Haar measure of $G/H$?

That is, is it true that $$\nu(A)=\mu(\pi^{-1}(A))$$ for every Borel $A\subseteq G/H$, where $\nu$ is the normalised Haar measure on $G/H$?

EDIT: I understand the pushforward measure is a left-invariant Borel measure assigning finite measure to compact sets and positive measure for open sets; but for it to be a Haar measure it also needs to be regular, at least in order to use the uniqueness of Haar measures proved in Hewitt and Ross's book (which is the only reference I know).

The answer is yes: if you write $$\pi^{-1}(\pi(A))=AH=HA$$ (since the subgroup $$H$$ is normal in $$G$$), then you realize that for any element $$gH=Hg\in G/H$$ you have $$gH.\pi(A)=gH.AH=g.HA=g.\pi^{-1}(\pi(A)).$$
Since the measure $$\mu$$ is invariant under the multiplication by $$g$$, so is the pushforward measure: $$\nu(gH.\pi(A))=\mu\left(\pi^{-1}[gH.\pi(A)]\right)=\mu(gHA)=\mu(AH)=\mu(\pi^{-1}[\pi(A)])=\nu(\pi(A)).$$
Moreover the pushforward measure is clearly non-trivial ($$\nu(G/H)=\mu(G)\neq 0$$), and Radon: (EDIT) by the definition of the quotient topology, the compact subsets of $$G/H$$ are of the form $$\pi(K)=KH$$, with $$K\subset G$$ compact. This implies that the measure $$\nu=\pi_*\mu$$ is inner regular.
• Thank you very much! That is a very good book. So one can either prove directly that radon measures are regular (in compact spaces) as in Proposition 1.1 here: www.math.uiuc.edu/~mjunge/54115/arv.pdf (and then use the uniqueness proved in Halmos's book), or one can even use another result in this book which says every left-invariant Borel measure (which is finite on compact sets and positive on open sets) is regular (at least in $\sigma$-compact groups, since his definition of Borel is a bit different than the standard). Your help is very appreciated! Nov 2, 2015 at 19:02