Concurrency, sigma-algebras, and probabilistic fairness

We extend previous constructions of probabilities for a prime event structure by allowing arbitrary confusion. Our study builds on results related to fairness in event structures that are of interest per se. Executions are captured by the set of maximal configurations. We show that the information collected by observing only fair executions is confined in some sigma-algebra contained in the Borel sigma-algebra. Equality holds when confusion is finite, but inclusion is strict in general. We show the existence of an increasing chain of sub-sigma-algebras that capture the information collected when observing executions of increasing unfairness. We show that, if the event structure unfolds a safe net, then unfairness remains quantitatively bounded, that is, the above chain gets steady in finitely many steps. The construction of probabilities typically relies on a Kolmogorov extension argument. We prove that, when the event structure unfolds a safe net, then unfair executions all belong to some set of zero probability. This yields a new construction of Markovian probabilistic nets, carrying a natural interpretation that ``unfair executions possess zero probability''