Under what conditions is there a strictly positive hyperreal probability measure on a group $G$? This would be a finitely-additive non-negative function $\mu$ from the powerset of $G$ to a hyperreal field $^\*R$ such that $\mu(A)>0$ for all $A\ne\varnothing$, $\mu(G)=1$ and $\mu(gA)=\mu(A)$ for all $g\in G$ and $A\subseteq G$.


A necessary condition is supramenability of $G$ (for non-empty $A$, the standard part of $\mu(\cdot)/\mu(A)$ will be an invariant measure on $G$) and $G$ being a torsion group (if $g$ has infinite order, let $A=\{e,g,g^2,...\}$, and strict positivity won't allow $\mu(gA)=\mu(A)$). Another necessary condition is that no subset of $G$ be equidecomposable with a proper subset of itself. (I don't know if this is stronger than the conjunction of the previous two conditions.)


I think a sufficient condition is that $G$ is locally finite (has no infinite finitely generated subgroups). **Sketch**: For any finite subgroup $H$ of $G$ and any finite boolean algebra of subsets of $G$ invariant under $H$ there is a strictly positive $H$-invariant finitely additive real probability measure; now use a fine ultrafilter on the poset of pairs of finite subgroups and finite boolean algebras.


It would be really neat if supramenability+torsion were sufficient. I guess a test case is the Grigorchuk group, but I don't know how to proceed there.



Invariant strictly positive hyperreal probability measures on groups
Local finiteness is not only sufficient but necessary for the existence of the requisite hyperreal measure. Not having a subset equidecomposable with a proper subset is clearly necessary. But this implies local finiteness [by a theorem of Scarparo](https://arxiv.org/pdf/1606.08262.pdf). In particular, supramenability+torsion is not sufficient, and there is no invariant strictly positive hyperreal probability measure on the Grigorchuk group.


