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?
Best Answer
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.