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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).

Cronus
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1 Answers1

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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.

Concerning the reference that you are asking, I really love the book by Halmos on Measure Theory. Another place where I've first learned about integration on locally compact groups is the appendices of the book by Bekka, de la Harpe & Valette about Kazhdan's property (T) (it is freely available online).

Amirhossein Haddadian
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user47274
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  • Why is this measure Radon? Also, could you please either (1) explain why this means the measure is regular or (2) give a reference for uniqueness of left-invariant Borel radon measures? – Cronus Nov 02 '15 at 13:45
  • 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! – Cronus Nov 02 '15 at 19:02
  • @Cronus. can we say a semilar statement for quotient haar measure for general locally compact group? – Amirhossein Haddadian Mar 21 '23 at 22:14