Concentration-compactness for Radon measures on a metric space

Question

It is known (see Ch. 4 in Struwe's Variational Methods) that Radon measures on $\mathbb{R}^n$ satisfy the concentration-compactness principle. Does the same hold true for Radon measures on a general metric space?

Answer 1

$\newcommand{\bR}{\mathbb{R}}$ The concentration compactness principle as P.L. Lions first formulated (roughly) states that a sequence of probability measures on $\mathbb{R}^n$ absolutely continuous with respect to the Lebesgue measure, fails to be relatively compact only due to the actions of the noncompact group of homotheties, i.e. transformations of the form $$ T_{x_0, t}:\bR^n\to\bR^n,\;\; x_0\in\bR^n$, t\in \bR, $$ $$ T_{x_0,t}(x)=x_0+tx,\;\;\forall x\in\bR^n. $$ Equivalently the space of orbits of the action of this group on the set probability measures is compact.

This phenomenon can be seen easily n by looking at the family of gaussian measures on $\bR$. They are determined by two parameters, the mean, the center of the Gauss bell, and the variance, the "width" of the Gauss bell. In fact trhe family of Gaussian measures is a single orbit of the action of the above group.

The orbit is non-compact. A sequence of Gaussian measures is non-compact if either the means go to infinity (visualize the energy bump under the bell running towards infinity) or the variance going to infinity (visualize a bell flattening so the energy is spread out over larger and larger regions). If the variances go to $0$ then the measures concentrate to Dirac measures, not absolutely continuous with respect to the Lebesgue measure.

On arbitrary Polish spaces (separable complete metric spaces) you have Prokhorov's compactness theorem. Lions' concentration-compactness principle is a direct application of this theorem.