# Invariant strictly positive hyperreal probability measures on groups

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.

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