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Let $P(\mathbb R)$ be the space of probability measures on $\mathbb R$ endowed with the Lévy Prokhorov metric. I know that it is a complete Polish space, but it is not Locally compact.

I wonder whether it is sigma compact or not (my intuition says it isn't).

Sadly metrizable, separable, complete and sigma compact do not imply locally compact.

Any idea?

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1 Answer 1

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It actually follows almost formally from the properties you have stated that $P(\mathbb{R})$ is not $\sigma$-compact. To be precise, you need a slightly stronger version of the fact that it is not locally compact: any nonempty complete metric space $X$ that is nowhere locally compact (i.e., no compact set has nonempty interior, or equivalently no closed ball is compact) is not $\sigma$-compact. Indeed, if $X$ were a countable union of compact subsets, then by the Baire category theorem one of those subsets would have to have nonempty interior. This is impossible, and so $X$ is not $\sigma$-compact.

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