Suppose we are given a locally compact topological space $X$ and a discreet group $G$ acting on it (we can assume the action to be proper). Given a Radon probability measure on the quotient space $G \backslash X$, how can we define a $G$-invariant measure on $X$. In particular, I am not sure how one shows that the preimage of Borel sets in $G \backslash X$ gives all $G$ invariant Borel sets in $X$. How about if $X$ is a Polish space ? Any reference or help would be appreciated.
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$\begingroup$ Given a Radon measure $\mu$ on $G \backslash X$, what if we just define $\psi(f) := \int_{G \backslash X} \sum_{x \in Gy} f(x) \ \text{d} \mu(y)$? This should define a $G$-equivariant positive linear functional on $\mathcal{C}_c (G)$, from which we conclude by the Riesz-Markov-Kakutani representation theorem. Is something missing? $\endgroup$– D. ThomineJun 2, 2021 at 9:46
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