This is a very technical note. I've spent a fair amount of time this week thinking about invariant Popper functions. Say that a group G is neatly supramenable if there is a Popper function P on G with every non-empty subset normal and satisfying the strong invariance condition P(gA|B)=P(A|B) whenever gA∪A⊆B. Neatly supramenable groups are supramenable: every non-empty subset A has an invariant finitely additive measure m such that m(A)=1. Anyway, I think I can prove—I now have two proofs drafted, so that makes me more confident—that every exponentially bounded group is neatly supramenable. Thus, every elementary supramenable group is neatly supramenable.
One philosophically interesting upshot of all this is that n-dimensional Euclidean space supports a Popper function with all non-empty subsets normal that is invariant under all translations as well as under single-coordinate reflections ((x1,...,xi,...,xn) going to (x1,...,−xi,...,xn)). But when one adds rotations into the mix, this is false for n≥2. So there is something philosophically problematic about rotations for the notion of uniform probability.
Don't quote the result yet as the proofs use mathematics that I am not very familiar with (ultrafilters, non-standard analysis, etc.).
Oh, and all of this uses the Axiom of Choice.
Turns out that Armstrong in a paper in this book has shown that (in my terminology) all supramenable groups are neatly supramenable.
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