Measure and category approximations for 𝐶-sets

Srivatsa, V. V.

doi:10.1090/S0002-9947-1983-0701507-4

Measure and category approximations for $C$-sets
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Trans. Amer. Math. Soc. 278 (1983), 495-505 Request permission

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

Effective selections and parametrizations

Dimitri P. Bertsekas and Steven E. Shreve, Stochastic optimal control, Mathematics in Science and Engineering, vol. 139, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. The discrete time case. MR 511544
D. Blackwell and C. Ryll-Nardzewski, Non-existence of everywhere proper conditional distributions, Ann. Math. Statist. 34 (1963), 223–225. MR 148097, DOI 10.1214/aoms/1177704259

Classical hierarchies from a modern standpoint, Part

sets

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
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
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
B. V. Rao, Remarks on analytic sets, Fund. Math. 66 (1969/70), 237–239. MR 274689, DOI 10.4064/fm-66-2-237-239
H. Sarbadhikari, Some uniformization results, Fund. Math. 97 (1977), no. 3, 209–214. MR 494012, DOI 10.4064/fm-97-3-209-214

Ob odnom klasse effectivnikh

35

Maurice Sion, Topological and measure-theoretic properties of analytic sets, Proc. Amer. Math. Soc. 11 (1960), 769–776. MR 131509, DOI 10.1090/S0002-9939-1960-0131509-0
V. V. Srivatsa, Existence of measurable selectors and parametrizations for $G_{\delta }$-valued multifunctions, Fund. Math. 122 (1984), no. 1, 23–32. MR 753010, DOI 10.4064/fm-122-1-23-32
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