nLab ergodic decomposition theorem

1. Idea

The ergodic decomposition theorem says that, under some conditions, every invariant measure is a mixture of ergodic ones.

In terms of category theory, the statement can be often expressed in terms of a categorical limit.

2. Statement of the theorem

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3. Examples

• De Finetti's theorem together with the Hewitt-Savage zero-one law can be seen as an instance of the ergodic decomposition theorem, where the action is given by finite permutations.

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4. Almost sure version

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5. Related entries

• probability theory, categorical probability
• invariant measure, invariant set
• ergodicity, ergodic theorem?
• de Finetti's theorem, zero-one law

6. References

• Terence Tao, What’s new? Lecture 9: Ergodicity, blog entry.

• Sean Moss, Paolo Perrone, A category-theoretic proof of the ergodic decomposition theorem, Ergodic Theory and Dynamical Systems, 2023. (arXiv:2207.07353)

• Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

category: probability