Principal bundle structure of the space of metric measure spaces

Abstract: We study the topological structure of the space $\mathcal{X}$ of isomorphism classes of metric measure spaces equipped with the box or concentration topologies. We consider the scale-change action of the multiplicative group $\mathbb{R}+$ of positive real numbers on $\mathcal{X}$, which has a one-point metric measure space, say $*$, as only one fixed-point. We prove that the $\mathbb{R}+$-action on $\mathcal{X}* := \mathcal{X} \setminus {*}$ admits the structure of nontrivial and locally trivial principal $\mathbb{R}+$-bundle over the quotient space. Our bundle $\mathbb{R}+ \to \mathcal{X}* \to \mathcal{X}*/\mathbb{R}+$ is a curious example of a nontrivial principal fiber bundle with contractible fiber. A similar statement is obtained for the pyramidal compactification of $\mathcal{X}$, where we completely determine the structure of the fixed-point set of the $\mathbb{R}_+$-action on the compactification.