A non-de Finetti theorem for countable Euclidean spaces

The classical de Finetti Theorem classifies the $\mathrm{S}\mathrm{y}\mathrm{m}(\mathbb{N})$-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 $\mathrm{Aut}(M)$ is de Finetti, i.e. $\mathrm{Aut}(M)$-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.