Symmetric qualitative (and other) probabilities

I recently worked out the precise conditions under which one can have Popper functions, hyperreal probabilities or qualitative probabilities that are invariant under some group of symmetries and are regular in the sense that they assign a bigger probability to non-empty sets than to the empty set.

But what if we don’t require regularity? Then the following is mainly a matter of putting together known theorems:

Proposition. Suppose G is a group acting on set Ω^* ⊇ Ω where Ω is non-empty. Then the following are equivalent:

1. There is a finitely additive G-invariant real-valued probability measure on the powerset of Ω

2. There is a finitely additive G-invariant hyperreal probability measure on the powerset of Ω

3. There is a finitely additive approximately G-invariant hyperreal probability measure on the powerset of Ω

4. There is a strongly G-invariant total qualitative probability ⪅ on the powerset of Ω such that ⌀ < Ω

5. There is a strongly G-invariant partial qualitative probability ⪅ on the powerset of Ω such that ⌀ < Ω

6. The set Ω is not G-paradoxical.

The definitions are in the paper I linked to at the top, except that approximate G-invariance only requires that P(A)−P(gA) be infinitesimal rather than requiring that it be zero.

Proof: Trivially, (a) implies (b) which implies (c). The standard part of a finitely additive approximately G-invariant hyperreal probability measure will be a finitely additive G-invariant real-valued probability measure, so (c) implies (a). Thus, (a)–(c) are equivalent.

Condition (a) implies condition (d): just define A ⪅ B iff P(A)≤P(B) where P is the measure in (a). And (d) implies (e) trivially.

Now we show that not-(f) implies not-(e). Suppose Ω is G-paradoxical, so Ω has disjoint subsets A and B with partitions A₁, ..., A_m and B₁, ..., B_n respectively, and there are elements g₁, ..., g_m and h₁, ..., h_n of G such that g₁A₁, ..., g_mA_m and h₁B₁, ..., h_nB_n are each a partition of Ω. Then by a standard result on qualitative probabilities (use the proof of Krantz, et al., Lemma 5.3.1.2):

1. A = A₁ ∪ ... ∪ A_m ≈ g₁A₁ ∪ ... ∪ g_mA_m = Ω

2. B = B₁ ∪ ... ∪ B_n ≈ h₁B₁ ∪ ... ∪ h_nB_n = Ω.

Since ⌀ < Ω, we have ⌀ < A by (1). By the proof of Corollary 5.3.1.2 in Krantz, et al., we have B < Ω iff ⌀ < Ω − B. But A ⊆ Ω − B, and ⌀ < A, so indeed we must have B < Ω, which contradicts (2).

Finally, Tarski’s Theorem says that (f) implies (a). □

Note 1: The two results from Krantz et al. are given for total qualitative probabilities, but the proofs do not use totality. (In the linked paper, I didn’t notice that Krantz et al. are working with total qualitative probabilities, but fortunately all works out.)

Note 2: There is a pleasing direct construction of a partial qualitative probability satisfying (d). For each A ⊆ Ω, let [A] be the corresponding member of the equidecomposability type semigroup. Then define A ⪅ B providing there is a c in the semigroup such that [A]+c ≤ [B]+c. It turns out that the condition ⌀ < Ω is then equivalent to 2[Ω]≠[Ω], i.e., is equivalent to the non-paradoxicality of Ω under G.