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Permutation invariant boolean states

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Abstract

We give a self-contained proof of the following result: Finitely additive probability measures (also known as “states”) of the free boolean algebra \({\mathsf F}_\omega \) over the free generating set \(\{X_1,X_2,\ldots \}\) having the invariance property under finite permutations of the \(X_i\), coincide with states lying in the closure of the set of convex combinations of product states of \({\mathsf F}_\omega \) in the vector space \(\mathbb R^{{\mathsf F}_\omega }\) equipped with the product topology. De Finetti’s celebrated exchangeability theorem can be easily recovered from our proof.

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Acknowledgements

The author is grateful to the Referee for her/his insightful comments and corrections, which resulted in a substantially improved manuscript.

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  1. Department of Mathematics and Computer Science “Ulisse Dini”, University of Florence, Viale Morgagni 67/A, 50134, Florence, Italy

    Daniele Mundici

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  1. Daniele Mundici
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Correspondence to Daniele Mundici.

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Presented by K. A. Kearnes.

To Bruno de Finetti, in memoriam.

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Mundici, D. Permutation invariant boolean states. Algebra Univers. 85, 27 (2024). https://doi.org/10.1007/s00012-024-00859-3

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