Normally measures and probability spaces are defined over $\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of $\sigma$-algebras? (i.e. complements do not always exist, however finite meets and countable/arbitrary joins do exist). Would there be any significant difficulty or difference in developing a measure theory for frames? Is existence of complements essential for measure theory?
It is sometime argued that open sets in topology (think of frames as pointless topology) capture the intuitive notion of observable events (e.g. see Steven Vickers' book "Topology via Logic"). The complement of an observable event does not need to be observable event in general (e.g. consider "there exists a white crow", assume that the number of crows are practically infinite so one cannot check all of them and observe that this statement is false, however to affirm it we need to observe a single white crow). It seems natural to want to assign probability to observable events and have a more general notion of measures that doesn't need complement.
Apologies if my question is naive.