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 Dof non-negative real numbers, with 0 is an element of D. Moreover, ordered D-valued universal submeasures exist for all such D. By using the Kechris - Pestov - Todorcevic 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. (C) 2021 Elsevier B.V. All rights reserved.
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