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I think I can solve the following exercise if $X$ is assumed to be separable, otherwise I can't.

Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a topological space $Y$ such that $Y$ is the union of a countable sequence of compact sets $(K_n)$, and such that $\pi^{-1} (K_n)$ is compact for each $n$. Let $\mu$ be a regular Borel probability measure on $Y$. Define the space $\mathcal{M}_\mu (X)_1$ consisting of all Borel probability measures $\nu$ on $X$ such that $\pi_* \nu = \mu$. Show that $\mathcal{M}_\mu(X)_1$ is compact with respect to the weak-* topology.

Now if $X$ is assumed to be separable, so is $C_0(X)$ and the same proof as in Helly-Bray's theorem works (mutatis mutandis). What if it's not?

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Isn't $\pi_*$ a continuous application from the space of Borel probability measures on $X$ to the set of Borel probability measures on $Y$, so that $\mathcal{M}_\mu(X)_1=(\pi_*)^{-1} (\mu)$ is closed ($\{\mu\}$ is closed because the weak-* topology is Hausdorff). Then since by Alaoglu's theorem the closed ball is compact with respect to the weak-* topology, we have the conclusion? What's wrong with this idea? – timofei Aug 22 '12 at 20:47
Just out of curiosity, where did this question come from? – Quinn Culver Aug 22 '12 at 23:29
An exercise in Renato Feres' book about dynamics and semisimple Lie groups. – timofei Aug 23 '12 at 6:59

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