Borel complexity of the space of probability measures
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- by Abhijit Dasgupta PDF
- Proc. Amer. Math. Soc. 129 (2001), 2441-2443
Abstract:
Using a technique developed by Louveau and Saint Raymond, we find the complexity of the space of probability measures in the Borel hierarchy: if $X$ is any non-Polish Borel subspace of a Polish space, then $P(X)$, the space of probability Borel measures on $X$ with the weak topology, is always true ${\boldsymbol {\Pi }^{\boldsymbol {0}}_{\boldsymbol {\xi }}}$, where $\xi$ is the least ordinal such that $X$ is ${\boldsymbol {\Pi }^{\boldsymbol {0}}_{\boldsymbol {\xi }}}$.References
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- Alexander S. Kechris, Measure and category in effective descriptive set theory, Ann. Math. Logic 5 (1972/73), 337–384. MR 369072, DOI 10.1016/0003-4843(73)90012-0
- A. Louveau and J. Saint-Raymond, Borel classes and closed games: Wadge-type and Hurewicz-type results, Trans. Amer. Math. Soc. 304 (1987), no. 2, 431–467. MR 911079, DOI 10.1090/S0002-9947-1987-0911079-0
- Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 561709
- K. R. Parthasarathy, Probability measures on metric spaces, Probability and Mathematical Statistics, No. 3, Academic Press, Inc., New York-London, 1967. MR 0226684
- Steven E. Shreve, Probability measures and the $C$-sets of Selivanovskij, Pacific J. Math. 79 (1978), no. 1, 189–196. MR 526678, DOI 10.2140/pjm.1978.79.189
Additional Information
- Abhijit Dasgupta
- Email: takdoom@yahoo.com
- Received by editor(s): October 24, 1994
- Received by editor(s) in revised form: November 24, 1999
- Published electronically: January 23, 2001
- Additional Notes: Supported in part by NSF Grant # DMS-9214048.
- Communicated by: Andreas R. Blass
- © Copyright 2001 Abhijit Dasgupta, GNU GPL style copyleft
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2441-2443
- MSC (2000): Primary 03E15, 60B05; Secondary 28A05
- DOI: https://doi.org/10.1090/S0002-9939-01-05801-4
- MathSciNet review: 1823929