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 $ν^{\otimes\mathbb N}$ where $ν$ 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 $ν^{\otimes M}$ where $ν$ is a measure on $[0,1]$. In this note, we give an example of a non-de Finetti non-Archimedean group.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Theorem 3: JT
• Theorem 4
• Proposition 5
• proof
• Proposition 6
• Proposition 7
• proof
• Theorem 8