Non-singular and probability measure-preserving actions of infinite permutation groups
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866918188564873216 |
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| author | Tsankov, Todor |
| author_facet | Tsankov, Todor |
| contents | We prove two theorems in the ergodic theory of infinite permutation groups. First, generalizing a theorem of Nessonov for the infinite symmetric group, we show that every non-singular action of a non-archimedean, Roelcke precompact, Polish group on a measure space $(Ω, μ)$ admits an invariant $σ$-finite measure equivalent to $μ$. Second, we prove the following de Finetti type theorem: if $G \curvearrowright M$ is a primitive permutation group with no algebraicity verifying an additional uniformity assumption, which is automatically satisfied if $G$ is Roelcke precompact, then any $G$-invariant, ergodic probability measure on $Z^M$, where $Z$ is a Polish space, is a product measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04716 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-singular and probability measure-preserving actions of infinite permutation groups Tsankov, Todor Dynamical Systems Logic 37A40, 37A50 We prove two theorems in the ergodic theory of infinite permutation groups. First, generalizing a theorem of Nessonov for the infinite symmetric group, we show that every non-singular action of a non-archimedean, Roelcke precompact, Polish group on a measure space $(Ω, μ)$ admits an invariant $σ$-finite measure equivalent to $μ$. Second, we prove the following de Finetti type theorem: if $G \curvearrowright M$ is a primitive permutation group with no algebraicity verifying an additional uniformity assumption, which is automatically satisfied if $G$ is Roelcke precompact, then any $G$-invariant, ergodic probability measure on $Z^M$, where $Z$ is a Polish space, is a product measure. |
| title | Non-singular and probability measure-preserving actions of infinite permutation groups |
| topic | Dynamical Systems Logic 37A40, 37A50 |
| url | https://arxiv.org/abs/2411.04716 |