  We introduce the Solecki submeasure $\sigma(A)=\inf_F\sup_{x,y\in G}|F\cap
xAy|/|F|$ and its left and right modifications on a group $G$, and study the
interplay between the Solecki submeasure and the Haar measure on compact
topological groups. Also we show that the right Solecki density on a countable
amenable group coincides with the upper Banach density $d^*$ which allows us to
generalize some fundamental results of Bogoliuboff, Folner, Cotlar and
Ricabarra, Ellis and Keynes about difference sets and Jin, Beiglbock, Bergelson
and Fish about the sumsets to the class of all amenable groups.
