equation involving the integral of the modular function of a topological group

Let $G$ be a locally compact topological group and $H$ a closed subgroup. Choose a left Haar measure $d\zeta$ for $H$, and let $d\mu$ be any measure for $G$. Also let $f$ and $g$ be continuous compactly supported real functions on $G$. I’m stumped by a step in a proof where the following equality is asserted :

$$\int_H \Delta_H(\zeta^{-1}) \Bigg[\int_G f(x\zeta^{-1})g(x)d\mu(x)\Bigg]d\zeta = \int_G g(x)\Bigg[ \int_H f(x\zeta) d\zeta\Bigg] d\mu (x).$$

I can only make the left hand side look like $$\int_G g(x) \Bigg[\int_H \Delta_H(\zeta^{-1})f(x\zeta^{-1})d\zeta \Bigg]d\mu (x)$$

Clearly I’m supposed to use the properties of the modular function $\Delta_H$, but I don’t know what to do when the argument for $\Delta_H$ is the variable I’m integrating over. Any help would be greatly appreciated. Thanks!

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Define a measure $d\zeta^{-1}$ on $H$ by the condition
$$
\int f(\zeta)\,d\zeta^{-1}=\int f(\zeta^{-1})\,d\zeta.
$$
I claim that there is an equality of measures
$$
\Delta_H(\zeta)\,d\zeta^{-1} = d\zeta.
$$
Integrating $f(x\zeta)$ against both these measures shows the inner integral on the right hand side of the original equality is equal to the inner integral in your transformation of the left hand side.

To prove the equality of measures: first one checks that $\Delta_H(\zeta)\,d\zeta^{-1}$ is a left Haar measure (this is basically equivalent to Lucien’s observation that $\Delta_H(\zeta^{-1})\,d\zeta$ is a right Haar measure). This means there is a constant $C>0$ such that $\Delta_H(\zeta)\,d\zeta^{-1}=C\,d\zeta$. Making the substitution $\zeta\leftrightarrow\zeta^{-1}$ shows that $C^2=1$, so that $C=1$.

Let us call $d\lambda(\zeta)$ your left Haar measure on $H$. Notice that, as already pointed out by Lucien, $\Delta_H(\zeta^{-1})d\lambda(\zeta)$ is a right Haar measure on $H$. Actually, it is exactly the measure $d\lambda(\zeta^{-1})$ (e.g. see Folland, “A course in abstract harmonic analysis”, p.48), i.e. the measure $\rho$ defined by $\rho(E)=\lambda(E^{-1})$. So your left hand side square bracket becomes

$$\int_H f(x\zeta^{-1})d\lambda(\zeta^{-1})$$

Now apply the general formula for the push-forward measure under the map $(\cdot)^{-1}\colon H\to H$ to get

$$\int_H f(x\zeta^{-1})d\lambda(\zeta^{-1})=\int_H f(x(\zeta^{-1})^{-1})d\lambda (\zeta)=\int_H f(x\zeta)d\lambda(\zeta)$$
and we’re done.