T C C N P  F

We define when a linear-time temporal property is a fairness property with respect to a given system. This captures the essence that is shared by most fairness assumptions that are used in the specification and verification of reactive concurrent systems, such as weak fairness, strong fairness, kfairness, and many others. We give three characterisations for the family of all fairness properties: a language-theoretic, a topological, and a gametheoretic characterisation. It turns out that the fairness properties are the “large” sets from a topological point of view, i.e., they are the co-meager sets in the natural topology of runs of a given system. This insight provides a link to probability theory where a set is “large” when it has measure 1. While these two notions of largeness are very similar, they do not coincide in general. However, we show that they coincide for ω-regular properties and bounded Borel measures. That is, an ω-regular temporal property of a finite-state system has measure 1 under a bounded Borel measure if and only if it is a fairness property with respect to that system.