Garrido, Definition 3.9 as printed — with measures valued in [0, 1], only the trivial group is supramenable

For a group $G$, the following are equivalent: for every nonempty $A \subseteq G$ there is a finitely additive, left-invariant $m : \mathcal P(G) \to [0, \infty]$ with $m(s) \le 1$ for every $s \subseteq G$ and $m(A) = 1$; and $G$ has at most one element.

IsFinitelyAdditiveMeasure ($m(\emptyset) = 0$ and $m(s \cup t) = m(s) + m(t)$ for disjoint $s, t$) and IsInvariant G m ($m(g s) = m(s)$ for all $g$ and $s$) are from the Garrido amenability definitions bundle; the bound $m(s) \le 1$ is the codomain $[0, 1]$ of the printed definition, written inside $[0, \infty]$.

Garrido writes on p. 11: “Definition 3.9. A group $G$ is supramenable if for every $\emptyset \neq A \subseteq G$ there is a finitely additive left-invariant measure $\mu : \mathcal P(G) \to [0, 1]$ such that $\mu(A) = 1$.” Read literally this is satisfied only by the trivial group: taking $A = \{1\}$, invariance gives $\mu(\{g\}) = 1$ for every $g$, and two distinct points would give $\mu(\{1, g\}) = 2$. The bundle's IsSupramenable therefore takes values in $[0, \infty]$, as in Rosenblatt's definition, under which Theorem 3.10(2) (Garrido.isSupramenable_of_isExponentiallyBounded: finitely generated groups of subexponential growth are supramenable, so $\mathbb Z$ is) holds; this theorem records why the printed codomain cannot be the intended one.