The Solecki submeasures on groups

The {\em Solecki submeasure} $\sigma$ on a group $G$ is the invariant monotone subadditive function assigning to each subset $A\subset G$ the real number $\sigma(A)=\inf_\sup_{x,y\in G}|F\cap xAy|/|F|$ where the infimum is taken over all non-empty finite subsets $F$ of $G$. In this paper we study the properties of the Solecki submeasure and its left and right modifications on (topological)groups and establish an interplay between the Solecki submeasure and the Haar measure on a compact topological group $G$. In particular, we show that the Haar measure on a compact topological group is uniquely determined by the Solecki submeasure. More details can be found at http://arxiv.org/abs/1211.0717.