Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Measure and category approximations for $ C$-sets

Author: V. V. Srivatsa
Journal: Trans. Amer. Math. Soc. 278 (1983), 495-505
MSC: Primary 04A15; Secondary 03E15, 28A99, 54H05
MathSciNet review: 701507
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The class of $ C$-sets in a Polish space is the smallest $ \sigma $-field containing the Borel sets and closed under operation $ (\mathcal{A})$. In this article we show that any $ C$-set in the product of two Polish spaces can be approximated (in measure and category), uniformly over all sections, by sets generated by rectangles with one side a $ C$-set and the other a Borel set. Such a formulation unifies many results in the literature. In particular, our methods yield a simpler proof of a selection theorem for $ C$-sets with $ {G_\delta }$-sections due to Burgess [4].

References [Enhancements On Off] (What's this?)

  • [1] R. Barua and V. V. Srivatsa, Effective selections and parametrizations, preprint.
  • [2] D. P. Bertsekas and S. E. Shreve, Stochastic optimal control: the discrete time case, Academic Press, New York, 1978. MR 511544 (80d:93081)
  • [3] D. Blackwell and C. Ryll-Nardzewski, Non-existence of everywhere proper conditional distributions, Ann. Math. Statist. 34 (1963), 223-225. MR 0148097 (26:5606)
  • [4] J. P. Burgess, Classical hierarchies from a modern standpoint, Part I: $ C$-sets, Fund. Math. (to appear).
  • [5] A. S. Kechris, Measure and category in effective descriptive set theory, Ann. Math. Logic 5 (1973), 337-384. MR 0369072 (51:5308)
  • [6] K. Kuratowski, Topology, Vol. 1, 5th ed., Academic Press, New York, 1966. MR 0217751 (36:840)
  • [7] Y. N. Moschovakis, Descriptive set theory, North-Holland, Amsterdam, 1980. MR 561709 (82e:03002)
  • [8] B. V. Rao, Remarks on analytic sets, Fund. Math. 66 (1970), 237-239. MR 0274689 (43:451)
  • [9] H. Sarbadhikari, Some uniformization results, Fund. Math 97 (1977), 209-214. MR 0494012 (58:12954)
  • [10] E. A. Selivanovskii, Ob odnom klasse effectivnikh, Mat. Sb. 35 (1928), 379-413.
  • [11] M. Sion, Topological and measure theoretic properties of analytic sets, Proc. Amer. Math. Soc. 11 (1960), 769-776. MR 0131509 (24:A1359)
  • [12] V. V. Srivatsa, Existence of measurable selectors and parametrizations for $ {G_\delta }$-valued multifunctions, Fund. Math. (to appear). MR 753010 (85k:28015)
  • [13] R. L. Vaught, Invariant sets in topology and logic, Fund. Math. 82 (1974), 269-294. MR 0363912 (51:167)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 04A15, 03E15, 28A99, 54H05

Retrieve articles in all journals with MSC: 04A15, 03E15, 28A99, 54H05

Additional Information

Keywords: Analytic set, $ C$-set, selections
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society