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The theory of random Borel sets as presented in part I of this paper is developed further. Special attention is payed to the reconstruction of the topology of the underlying space from our presentation of the measure algebra, to an analysis of capacities in context of random Borel sets, to inspection processes on the unit segment and to the Markov process of random allocation. 2010 MSC: Primary 60D05; Secondary 54H05 Proposition 2.1. If a type I resolution is used for the isomorphism theoremdoi:10.4995/agt.2012.1639 fatcat:2acedj73grf43cd4ykid4wp2eq