On the automorphism groups of universal submeasures

Abstract

We investigate dynamical properties of the automorphism groups of general versions of the universal submeasures defined in [3]. First, we show that, a universal submeasure D-valued exists for every countable (finite or infinite) set D of non-negative real numbers, with 0∈D. Moreover, ordered D-valued universal submeasures exist for all such D. By using the Kechris - Pestov - Todorčević theory, we prove that for all the ordered universal submeasures, automorphism groups are extremely amenable, but they do not have ample generics when D satisfies some additional conditions. Finally, we prove that the class of all finite D-valued submeasures has the Hrushovski property, and the automorphism group of the D-valued universal submeasure is amenable and has ample generics.