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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.

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  • $\begingroup$ Standard terminology for "$G$ has no infinite finitely generated subgroup" is "$G$ is locally finite". $\endgroup$
    – YCor
    Aug 25, 2020 at 14:08
  • $\begingroup$ I was searching yesterday for the term but couldn't find it! Thanks! $\endgroup$ Aug 25, 2020 at 14:14

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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. In particular, supramenability+torsion is not sufficient, and there is no invariant strictly positive hyperreal probability measure on the Grigorchuk group.

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