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Let $G$ be a locally compact group with left Haar measure $\mu$. Furthermore, $\Gamma_1, \Gamma_2 \subset G$ are such that $\mu$ induces finite Haar measures $\mu_1$ and $\mu_2$ on $G /\Gamma_1$ and $G/ \Gamma_2$, respectively ($\Gamma_1$ and $\Gamma_2$ are not necessarily discrete as in the case of a lattice).

Then, for $i=1,2$, define densities $\delta_i (E) = \frac{\mu_i(E)}{\mu_i(G/\Gamma_i)}$ for measurable sets $E \subset G$ with the property that $E = E\Gamma_1 = E\Gamma_2$ (then $E \subset G/\Gamma_i$ makes sense).

Now I try to prove that $\delta_1 (E) = \delta_2(E)$. Is this a known result, and if so, where could I find it?

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