In many interesting cases, there is no way to define a regular hyperreal-valued probability that is invariant under symmetries, where “regular” means that every non-empty set has non-zero probability. For instance, there is no such measure for all subsets of the circle with respect to rotations: the best we can do is approximate invariance, where *P*(*A*)−*P*(*r**A*) is infinitesimal for every rotation. On the other hand, I have recently shown that there is such a measure for infinite sequences of fair coin tosses where the symmetries are reversals at a set of locations.

So, here’s an interesting question: Given a space *Ω* and a group *G* of symmetries acting on *Ω*, under what exact conditions is there a hyperreal finitely-additive probability measure *P* defined for all subsets of *Ω* that satisfies the regularity condition *P*(*A*)>0 for all non-empty *A* and yet is fully (and not merely approximately) invariant under *G*, so that *P*(*g**A*)=*P*(*A*) for all *g* ∈ *G* and *A* ⊆ *Ω*?

**Theorem:** Such a measure exists if and only if the action of *G* on *Ω* is locally finite. (Assuming the Axiom of Choice.)

The action of *G* on *Ω* is locally finite iff for every *x* ∈ *Ω* and every finitely-generated subgroup *H* of *G*, the orbit *H**x* = {*h**x* : *h* ∈ *H*} of *x* under *H* is finite. In other words, we have such a measure provided that applying the symmetries to any point of the space only generates finitely many points.

This mathematical fact leads to a philosophical question: Is there anything *philosophically* interesting about those symmetries whose action is locally finite? But I’ve spent so much of the day thinking about the mathematical question that I am too tired to think very hard about the philosophical question.

**Sketch of Proof of Theorem:** If some subset *A* of *Ω* is equidecomposable with a proper subset *A*′, then a *G*-invariant measure *P* will assign equal measure to both *A* and *A*′, and hence will assign zero measure to the non-empty set *A* − *A*′, violating the regularity condition. So, if the requisite measure exists, no subset is equidecomposable with a proper subset of itself, which by a theorem of Scarparo implies that the action of *G* is locally finite.

Now for the converse. If we could show the result for all finitely-generated groups *G*, by using ultraproduct along an ultrafilter on the partially ordered set of all finitely generated subgroups of *G* we could show this for a general *G*.

So, suppose that *G* is finitely generated and the orbit of *x* under *G* is finite for all *x* ∈ *Ω*. A subset *A* of *G* is said to be *G*-invariant provided that *g**A* = *A* for all *g* ∈ *G*. The orbit of *x* under *G* is always *G*-invariant, and hence every finite subset of *A* is contained in a finite *G*-invariant subset, namely the union of the orbits of all the points in *A*.

Consider the set *F* of all finite *G*-invariant subsets of *Ω*. It’s worth noting that every finite subset of *G* is contained in a finite *G*-closed subset: just take the union of the orbits under *G*. For *A* ∈ *F*, let *P*_{A} be uniform measure on *A*. Let *F*^{*} = {{*B* ∈ *F* : *A* ⊆ *B*}:*A* ∈ *F*}. This is a non-empty set with the finite intersection property. Let *U* be an ultrafilter extending *F*^{*}. Let ^{*}*R* be the ultraproduct of the reals over *F* with respect to *U*, and let *P*(*C*) be the equivalence class of the function *A* ↦ *P*_{A}(*A* ∩ *C*) on *F*. Note that *C* ↦ *P*_{A}(*A* ∩ *C*) is *G*-invariant for any *G*-invariant set *A*, so *P* is *G*-invariant. Moreover, *P*(*C*)>0 if *C* ≠ ∅. For let *C*′ be the orbit of some element of *C*. Then {*B* ∈ *F* : *C*′⊆*B*} is in *F*^{*}, and *P*_{A}(*A* ∩ *C*′) > 0 for all *A* such that *C*′⊆*A*, so the set of all *A* such that *P*_{A}(*A* ∩ *C*′) > 0 is in *U*. It follows that *P*(*C*′) > 0. But *C*′ is the orbit of some element *x* of *C*, so every singleton subset of *C*′ has the same *P*-measure as {*x*} by the *G*-invariance of *P*. So *P*({*x*}) = *P*(*C*′)/|*C*′| > 0, and hence *P*(*C*)≥*P*({*x*}) > 0.

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