Borel functions of bounded class
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- by D. H. Fremlin, R. W. Hansell and H. J. K. Junnila PDF
- Trans. Amer. Math. Soc. 277 (1983), 835-849 Request permission
Abstract:
Let $X$ and $Y$ be metric spaces and $f:X \to Y$ a Borel measurable function. Does $f$ have to be of bounded class, i.e. are the sets ${f^{ - 1}}[ H ]$, for open $H \subseteq Y$, of bounded Baire class in $X?$ This is an old problem of A. H. Stone. Positive answers have been given under a variety of extra hypotheses and special axioms. Here we show that (i) unless something similar to a measurable cardinal exists, then $f$ is of bounded class and (ii) if $f$ is actually a Borel isomorphism, then $f ({\text {and}}\ {f^{ - 1}})$ are of bounded class.References
- R. H. Bing, W. W. Bledsoe, and R. D. Mauldin, Sets generated by rectangles, Pacific J. Math. 51 (1974), 27–36. MR 357124
- W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Die Grundlehren der mathematischen Wissenschaften, Band 211, Springer-Verlag, New York-Heidelberg, 1974. MR 0396267 F. Drake, Set theory, North-Holland, Amsterdam, 1974.
- William G. Fleissner, An axiom for nonseparable Borel theory, Trans. Amer. Math. Soc. 251 (1979), 309–328. MR 531982, DOI 10.1090/S0002-9947-1979-0531982-9
- William G. Fleissner, Roger W. Hansell, and Heikki J. K. Junnila, PMEA implies proposition $\textrm {P}$, Topology Appl. 13 (1982), no. 3, 255–262. MR 651508, DOI 10.1016/0166-8641(82)90034-7
- R. W. Hansell, Borel-additive families and Borel maps in metric spaces, General topology and modern analysis (Proc. Conf., Univ. California, Riverside, Calif., 1980) Academic Press, New York-London, 1981, pp. 405–416. MR 619067
- R. W. Hansell, Point-finite Borel-additive families are of bounded class, Proc. Amer. Math. Soc. 83 (1981), no. 2, 375–378. MR 624935, DOI 10.1090/S0002-9939-1981-0624935-8
- J. Kaniewski and R. Pol, Borel-measurable selectors for compact-valued mappings in the non-separable case, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), no. 10, 1043–1050 (English, with Russian summary). MR 410657 K. Kunen, Inaccessibility properties of cardinals, Ph. D. dissertation, Stanford Univ., 1968.
- 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
- D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. Math. Logic 2 (1970), no. 2, 143–178. MR 270904, DOI 10.1016/0003-4843(70)90009-4
- Arnold W. Miller, On the length of Borel hierarchies, Ann. Math. Logic 16 (1979), no. 3, 233–267. MR 548475, DOI 10.1016/0003-4843(79)90003-2
- David Preiss, Completely additive disjoint system of Baire sets is of bounded class, Comment. Math. Univ. Carolinae 15 (1974), 341–344. MR 346116
- B. V. Rao, On discrete Borel spaces and projective sets, Bull. Amer. Math. Soc. 75 (1969), 614–617. MR 242684, DOI 10.1090/S0002-9904-1969-12264-0
- Robert M. Solovay, Real-valued measurable cardinals, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 397–428. MR 0290961
- A. H. Stone, Non-separable Borel sets, Rozprawy Mat. 28 (1962), 41. MR 152457
- A. H. Stone, Some problems of measurability, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Lecture Notes in Math., Vol. 375, Springer, Berlin, 1974, pp. 242–248. MR 0364589 B. S. Spahn, Thesis, Warsaw, 1981.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 277 (1983), 835-849
- MSC: Primary 54H05; Secondary 03E15
- DOI: https://doi.org/10.1090/S0002-9947-1983-0694392-0
- MathSciNet review: 694392