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Tarski proved that if a group $G$ is exponentially bounded, then for $a$, $b$ and $c$ in the associated (equidecomposability) type semigroup, we have $a+c=b+2c \Rightarrow a=b+c$. Question: Can thi …

The Tarski condition $a+c=b+2c\Rightarrow a=b+c$ is equivalent to supramenability, given AC. Proof: First, note that the Tarski condition on the type space $S=S(G)/G$ is easily equivalent to the con …

Given a field $F$ of subsets of $\Omega$, we can define full conditional probabilities to be a function $P:F\times (F-\{ \varnothing \}) \to [0,1]$ such that: $P(-|B)$ is a finitely-additive probabi …

The answer is negative and the counterexample is a lot easier than the two cases I considered in the question. (It would still be nice to have an answer to the two cases.) Let $\Omega = \mathbb Z$, …

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 f …

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 imp …

The Sierpinski-Mazurkiewicz paradox yields a nonempty rigid-motion paradoxical subset $S$ of the Euclidean plane: $S$ is the disjoint union of $A$ and $B$, each of which is $G$-equidecomposable with $ …

Let $G = \mathbb Z_2^\omega$, with pointwise addition. Assume the Axiom of Choice. I am interested in finitely additive probability measures $\mu$ defined on all of $\mathcal PG$ that can be intuiti …

It's been a while since I've worked with amenable groups and integrals against finitely additive measures, so I could be missing something, but it now seems to me that the question is easy. While appa …

It looks like neat supramenability is equivalent to supramenability, at least given AC. This follows from Proposition 1.7 in the 1989 paper by Armstrong in this volume (page 7). The proof uses the ex …

Suppose a group $G$ acts on an infinite set $X$ and $X$ has no non-empty $G$-paradoxical subsets. Is it possible for $X$ to have non-trivial $G$-paradoxical subsets modulo finite sets? I.e., can there …

A group $G$ is supramenable iff for all $\varnothing\ne A\subseteq G$ there is a finitely-additive left-$G$-invariant measure $\mu_A$ on $G$ with $\mu_A(A)=1$. I'm interested in a seemingly stronger …