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One-to-one Borel selection theorems
Author(s):
Marek
Balcerzak;
Joanna
Peredko
Journal:
Proc. Amer. Math. Soc.
127
(1999),
2759-2766.
MSC (1991):
Primary 04A15, 28A05, 54H05
Posted:
April 23, 1999
MathSciNet review:
1487357
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Abstract:
For ![$X=[0,1]$](/proc/1999-127-09/S0002-9939-99-04784-X/gif-abstract/img1.gif) we obtain new theorems stating that a Borel set in  with large sets of large vertical and large horizontal sections admits a one-to-one Borel selection with large domain and large range. Largeness is meant mainly in measure or category sense. Our proofs combine a result of Graf and Mauldin with a modified result of Sarbadhikari.
References:
- [DSR]
- G. Debs, J. Saint Raymond, Séléctions Boréliennes injectives, Amer. J. Math., 111 (1989), 519-534. MR 90f:54060
- [GM]
- S. Graf, R.D. Mauldin, Measurable one-to-one selections and transition kernels, Amer. J. Math., 107 (1985), 407-425. MR 86e:54044
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- A.S. Kechris, Classical Descriptive Set Theory, Springer, New York 1995. MR 96e:03057
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- K. Kuratowski, Topology I, Academic Press, New York 1966. MR 36:840
- [M1]
- R.D. Mauldin, One-to-one selections-marriage theorems, Amer. J. Math., 104 (1982), 823-828. MR 84b:28013
- [M2]
- R.D. Mauldin, One-to-one selections and orthogonal transition kernels, Contemp. Math., 94 (1989), 185-190. MR 90h:28001
- [MS]
- R.D. Mauldin, H. Sarbadhikari, Continuous one-to-one parametrizations, Bull. Sc. Math.,
 série, 105 (1981), 435-444. MR 83i:28015 - [O]
- J.C. Oxtoby, Measure and Category, Springer, New York 1971. MR 52:14213
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- H. Sarbadhikari, A one-to-one selection theorem, Proc. Amer. Math. Soc., 97 (1986), 320-322. MR 87f:54031
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- D.H. Wagner, Survey of measurable selection theorems, Siam J. Control and Optimization, 15 (1977), 859-903. MR 58:6137
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Additional Information:
Marek
Balcerzak
Affiliation:
Institute of Mathematics, Lódz Technical University, al. Politechniki 11, 90-924 Lódz, Poland
Email:
mbalce@krysia.uni.lodz.pl
Joanna
Peredko
Affiliation:
Institute of Mathematics, Lódz Technical University, al. Politechniki 11, 90-924 Lódz, Poland
Email:
joannape@ck-sg.p.lodz.pl
DOI:
10.1090/S0002-9939-99-04784-X
PII:
S 0002-9939(99)04784-X
Keywords:
One-to-one selection,
Borel set,
meager set,
Lebesgue null set
Received by editor(s):
November 3, 1997
Posted:
April 23, 1999
Communicated by:
Frederick W. Gehring
Copyright of article:
Copyright
1999,
American Mathematical Society
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