I'm reading the paper Kesten’s theorem for Invariant Random Subgroups and am trying to understand what exactly is meant by labeling the vertices of the random variable $G$ using a uniform i.i.d. - in what way does this serve to create the measure $\mu$, which is supposed to end up being $\Gamma$-invariant. Also, I'm not very clear on how $X$ ends up being a Borel-space. What is the product topology referring to here (the Schreier graphs have the Chabauty topology, but what about the label functions?)
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