A non-de Finetti theorem for countable Euclidean spaces

Abstract: The classical de Finetti Theorem classifies the $\mathrm{Sym}(\mathbb N)$-invariant probability measures on $[0,1]^{\mathbb N}$. More precisely it states that those invariant measures are combinations of measures of the form $\nu^{\otimes\mathbb N}$ where $\nu$ is a measure on $[0,1]$. Recently, Jahel--Tsankov generalized this theorem showing that under conditions on $M$, the group $\operatorname{Aut}(M)$ is de Finetti, i.e. $\operatorname{Aut}(M)$-invariant measures on $[0,1]^M$ are mixtures of measures of the form $\nu^{\otimes M}$ where $\nu$ is a measure on $[0,1]$. In this note, we give an example of a non-de Finetti non-Archimedean group.