For probabilities that have values, regularity is normally defined as the claim that P(A)>0 whenever A is non-empty. Given finite additivity, this entails that if A is a proper subset of B, then P(A)<P(B). If instead of assigning probability values we deal with qualitative probabilities—i.e., probability comparisons—we now have a choice of how to define regularity:
- the probability comparison ≤ is weakly regular provided that ∅<A whenever A is non-empty
- the probability comparison ≤ is strongly regular provided that A<B whenever A is a proper subset of B.
If one assumes the axiom:
- (Additivity, de Finetti) If A∩C=∅ and B∩C=∅, then A≥B if and only if A∪C≥B∪C,
In a recent post, I showed that there is no rotationally invariant strongly regular qualitative probability defined on all countable subsets of the circle. But perhaps there is a useful weakly regular one that does not satisfy Additivity?
I don't know. Here's a start. Let P be the Bernstein-Wattenberg hyperreal-valued measure on the circle. Define A≤B if and only if for some rotations r and s we have P(rA)≤P(sB). Define A≤B if and only if not B<A. Then ≤ is weakly regular, and rotationally invariant. But I can't prove that it's transitive. Is it?