Problems for isometrically invariant Popper functions in the plane

Popper functions encode finitely-additive conditional probabilities. The hope of Popper functions is to allow conditionalizations on sets that classically have null probability, such as finite sets in a continuous context. A Popper function allows for non-trivial conditionalization on a set A only if A is normal, i.e., P(∅|A)=0. A Popper function P is weakly invariant under isometries (the group generated by rotations, translations and reflections, in the case of the plane) provided that P(gA|gB)=P(A|B) for any isometry g and A and B such that A,B,gA,gB are subsets of our probability space Ω. (Strong invariance would also say that P(gA|B)=P(A|B) and P(A|gB)=P(A|B) under appropriate circumstances.)

In an earlier post I basically proved (without the Axiom of Choice) that any weakly isometrically invariant Popper function on a solid three-dimensional ball that makes all countable sets measurable also makes all finite sets abnormal. This triviality result of course generalizes to higher dimensions.

I am now able to prove that if P is a weakly isometrically invariant Popper function on a subset Ω of the plane that contains a solid disc, with P defined at least for all countable sets, then there is a countable abnormal set. Again, since subsets of an abormal set are abnormal, it follows that some singleton is abnormal, and hence all singletons are abnormal by invariance, and hence all finite sets are abnormal as finite unions of abnormal sets are abnormal.

A proof is very roughly sketched here. It uses Just's construction of a bounded paradoxical subset of the plane (cf. this paper which gives another construction that could be used).

It also follows that there is no weakly isometrically invariant comparative probability function defined for all nonempty pairs of subsets of Ω (for definition of comparative probability, see my answer here).

On the other hand, Parikh and Parnes have shown that there is a Popper function defined for all pairs of subsets of the unit interval on the line making all nonempty subsets normal. Moreover I think their proof can be used to show that there is a translation-invariant Popper function defined for all pairs of subsets of n-dimensional Euclidean space making all non-empty subsets normal. We can get invariance under reflection in any finite set of orthogonal hyperplanes for free just by averaging combinations.

So the problem really is with rotations. Something important changes when one goes from dimension one to higher dimensions: rotations become available.