On extending actions
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- by Robert Vaught PDF
- Proc. Amer. Math. Soc. 107 (1989), 1087-1090 Request permission
Abstract:
Consider a Polish topological group $G$ acting via $J$ on a substandard (= countably generated) Borel space. Theorem 1. Any such "Borel action" can be extended to a Borel action $J’:G \times X’ \to X’$ where $X’$ is coanalytic. (Theorem 3 gives an analogue for continuous actions.) Corollary 2. The result "in any Borel action, orbits are Borel" implies the (well-known) result "all such orbits are absolutely Borel".References
- Casimir Kuratowski, Topologie. I. Espaces Métrisables, Espaces Complets, Monografie Matematyczne, Vol. 20, Państwowe Wydawnictwo Naukowe (PWN), Warszawa-Wrocław, 1948 (French). 2d ed. MR 0028007 —, Topology, Vol. 1, Panstwowe Wydawnictwo Naukowe (Warsaw) and Pergamon Press (New York, Oxford), 1966.
- Douglas E. Miller, On the measurability of orbits in Borel actions, Proc. Amer. Math. Soc. 63 (1977), no. 1, 165–170. MR 440519, DOI 10.1090/S0002-9939-1977-0440519-8
- Robert Vaught, Invariant sets in topology and logic, Fund. Math. 82 (1974/75), 269–294. MR 363912, DOI 10.4064/fm-82-3-269-294
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 107 (1989), 1087-1090
- MSC: Primary 54H05; Secondary 54H15
- DOI: https://doi.org/10.1090/S0002-9939-1989-0962248-2
- MathSciNet review: 962248