Let X be the set of integers, or a circle, or the real line, or Euclidean n dimensional space. Imagine a point is "uniformly" randomly chosen in X. For any two subsets A and B of X, we would like to be able to say if one of the subsets is more probable as the location of the point. Here are some conditions we want to impose on the ≤ comparison:
- ≤ is a total preorder: for any A, B and C, we have A≤B or B≤A; A≤B and B≤C implies A≤C; and A≤A.
- For any translation t of X (where we deem rotations on the circle to count as "circular translations"), we have tA≤tB if and only if A≤B. (≤ is translation-invariant)
- If A is a proper subset of B, then A<B (i.e., A≤B but not B≤A).
- If m(A)<m(B) for d-dimensional Hausdorff measure (including of course Lebesgue measure), for any d between 0 and the dimension of the space (inclusive), then A<B.
Proposition 1. Given the Axiom of Choice, there is an ordering ≤ satisfying 1-4.
Proof: Start with an ordering such that A is less than B if and only if A=B, or m(A)<m(B) for any d-dimensional Hausdorff measure, or A is a proper subset of B. This ordering is translation-invariant, and it extends to a preorder satisfying 1-4 by the main theorem of Section 2 here.
That sounds great! We can finally compare probabilities of landing in arbitrary sets, it seems. Well, almost. Given a uniform distribution, we would at least want the invariance also to hold for coordinate reflections (where we reflect the kth coordinate, for any k).
Proposition 2. There is no ordering ≤ satisfying 1-3 and the coordinate reflection condition.
That's a consequence of the final proposition in the paper I linked to above.
What a surprising difference these reflections make! With just translations, we have a lovely invariant order (though presumably not unique) respecting strict inclusions of sets. When we add coordinate reflections, we don't. Technically, the difference is that once we have reflections and translations, our symmetry group is no longer commutative. And of course, in the Euclidean space case if we add rotations, all is lost, too (that, too, is easy to show).
Philosophical corollary. There can be incommensurably probable events, and hence incommensurably valuable events (since two chances at the same good will be incommensurably good if the chances are incommensurably probable).