Is an arbitrary diffused Borel probability measure in a Polish space without isolated points Haar measure?

Abstract

It is introduced a certain approach for equipment of an arbitrary set of the cardinality of the continuum by structures of Polish groups and two-sided (left or right) invariant Haar measures. By using this approach we answer positively Maleki's certain question(2012) {\it what are the real $k$-dimensional manifolds with at least two different Lie group structures that have the same Haar measure.} It is demonstrated that for each diffused Borel probability measure $\mu$ defined in a Polish space $(G,\rho,\mathcal{B}_{\rho}(G))$ without isolated points there exist a metric $\rho_1$ and a group operation $\odot$ in $G$ such that $\mathcal{B}_{\rho}(G)=\mathcal{B}_{\rho_1}(G)$ and $(G,\rho_1, \mathcal{B}_{\rho_1}(G), \odot)$ stands a compact Polish group with a two-sided (left or right) invariant Haar measure $\mu$, where $\mathcal{B}_{\rho}(G)$ and $\mathcal{B}_{\rho_1}(G)$ denote Borel $\sigma$ algebras of subsets of $G$ generated by metrics $\rho$ and $\rho_1$, respectively. Similar result is obtained for construction of locally compact non-compact or non-locally compact Polish groups equipped with two-sided (left or right) invariant quasi-finite Borel measures.