When are Borel functions Baire functions?

M. Fosgerau

Fundamenta Mathematicae (1993)

  • Volume: 143, Issue: 2, page 137-152
  • ISSN: 0016-2736

Abstract

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The following two theorems give the flavour of what will be proved. Theorem. Let Y be a complete metric space. Then the families of first Baire class functions and of first Borel class functions from [0,1] to Y coincide if and only if Y is connected and locally connected.Theorem. Let Y be a separable metric space. Then the families of second Baire class functions and of second Borel class functions from [0,1] to Y coincide if and only if for all finite sequences U 1 , . . . , U q of nonempty open subsets of Y there exists a continuous function ϕ:[0,1] → Y such that ϕ - 1 ( U i ) ≠ Ø for all i ≤ q.

How to cite

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Fosgerau, M.. "When are Borel functions Baire functions?." Fundamenta Mathematicae 143.2 (1993): 137-152. <http://eudml.org/doc/211997>.

@article{Fosgerau1993,
abstract = {The following two theorems give the flavour of what will be proved. Theorem. Let Y be a complete metric space. Then the families of first Baire class functions and of first Borel class functions from [0,1] to Y coincide if and only if Y is connected and locally connected.Theorem. Let Y be a separable metric space. Then the families of second Baire class functions and of second Borel class functions from [0,1] to Y coincide if and only if for all finite sequences $U_1,...,U_q$ of nonempty open subsets of Y there exists a continuous function ϕ:[0,1] → Y such that $ ϕ^\{-1\}(U_i) ≠Ø$ for all i ≤ q. },
author = {Fosgerau, M.},
journal = {Fundamenta Mathematicae},
keywords = {Baire functions; Borel functions; functions of the first class},
language = {eng},
number = {2},
pages = {137-152},
title = {When are Borel functions Baire functions?},
url = {http://eudml.org/doc/211997},
volume = {143},
year = {1993},
}

TY - JOUR
AU - Fosgerau, M.
TI - When are Borel functions Baire functions?
JO - Fundamenta Mathematicae
PY - 1993
VL - 143
IS - 2
SP - 137
EP - 152
AB - The following two theorems give the flavour of what will be proved. Theorem. Let Y be a complete metric space. Then the families of first Baire class functions and of first Borel class functions from [0,1] to Y coincide if and only if Y is connected and locally connected.Theorem. Let Y be a separable metric space. Then the families of second Baire class functions and of second Borel class functions from [0,1] to Y coincide if and only if for all finite sequences $U_1,...,U_q$ of nonempty open subsets of Y there exists a continuous function ϕ:[0,1] → Y such that $ ϕ^{-1}(U_i) ≠Ø$ for all i ≤ q.
LA - eng
KW - Baire functions; Borel functions; functions of the first class
UR - http://eudml.org/doc/211997
ER -

References

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  1. [1] S. Banach, Oeuvres, Vol. 1, PWN, Warszawa, 1967, 207-217. 
  2. [2] L. G. Brown, Baire functions and extreme points, Amer. Math. Monthly 79 (1972), 1016-1018. Zbl0254.26008
  3. [3] R. Engelking, General Topology, PWN-Polish Scientific Publishers, Warszawa, 1977. 
  4. [4] W. G. Fleissner, An axiom for nonseparable Borel theory, Trans. Amer. Math. Soc. 251 (1979), 309-328. Zbl0428.03044
  5. [5] R. W. Hansell, Borel measurable mappings for nonseparable metric spaces, ibid. 161 (1971), 145-169. 
  6. [6] R. W. Hansell, On Borel mappings and Baire functions, ibid. 194 (1974), 195-211. 
  7. [7] R. W. Hansell, Extended Bochner measurable selectors, Math. Ann. 277 (1987), 79-94. Zbl0598.28019
  8. [8] R. W. Hansell, First class selectors for upper semi-continuous multifunctions, J. Funct. Anal. 75 (1987), 382-395. Zbl0644.54014
  9. [9] R. W. Hansell, First class functions with values in nonseparable spaces, in: Constantin Carathéodory: An International Tribute, T. M. Rassias (ed.), World Sci., Singapore, 1992, 461-475. 
  10. [10] K. Kuratowski, Topology, Vol. 1, Academic Press, New York, 1966. 
  11. [11] K. Kuratowski, Topology, Vol. 2, Academic Press, New York, 1968. 
  12. [12] M. Laczkowich, Baire 1 functions, Real Anal. Exchange 9 (1983-84), 15-28. 
  13. [13] C. A. Rogers, Functions of the first Baire class, J. London Math. Soc. (2) 37 (1988), 535-544. Zbl0687.54014
  14. [14] S. Rolewicz, On inversion of non-linear transformations, Studia Math. 17 (1958), 79-83. Zbl0086.10104

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