Decomposing Borel sets and functions and the structure of Baire Class 1 functions
Author:
Slawomir Solecki
Journal:
J. Amer. Math. Soc. 11 (1998), 521550
MSC (1991):
Primary 03A15, 26A21, 28A12
MathSciNet review:
1606843
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Abstract: We establish dichotomy results concerning the structure of Baire class 1 functions. We consider decompositions of Baire class 1 functions into continuous functions and into continuous functions with closed domains. Dichotomy results for both of them are proved: a Baire class 1 function decomposes into countably many countinuous functions, or else contains a function which turns out to be as complicated with respect to the decomposition as any other Baire class 1 function; similarly for decompositions into continuous functions with closed domains. These results strengthen a theorem of Jayne and Rogers and answer some questions of Steprans. Their proofs use effective descriptive set theory as well as infinite Borel games on the integers. An important role in the proofs is played by what we call, in analogy with being Wadge complete, complete semicontinuous functions. As another application of our study of complete semicontinuous functions, we generalize some recent theorems of Jackson and Mauldin, and van Mill and Pol concerning measures viewed as examples of complicated semicontinuous functions. We also prove that a Borel set is either or there is a continuous injection such that for any set , is meager. We show analogous results for Borel functions. These theorems give a new proof of a result of Stern, strengthen some results of Laczkovich, and improve the estimates for cardinal coefficients studied by Cichon, Morayne, Pawlikowski, and the author.
 [AN]
P.
S. Novikov and S.
I. Adyan, On a semicontinuous function, Moskov. Gos. Ped.
Inst. Uč. Zap. 138 (1958), 3–10 (Russian). MR 0120326
(22 #11081)
 [BD]
H. Becker and R. Dougherty, On disjoint Borel uniformizations, Adv. Math. (to appear).
 [CM]
J.
Cichoń and M.
Morayne, Universal functions and generalized
classes of functions, Proc. Amer. Math.
Soc. 102 (1988), no. 1, 83–89. MR 915721
(89c:26003), http://dx.doi.org/10.1090/S00029939198809157216
 [CMPS]
J.
Cichoń, M.
Morayne, J.
Pawlikowski, and S.
Solecki, Decomposing Baire functions, J. Symbolic Logic
56 (1991), no. 4, 1273–1283. MR 1136456
(92j:04001), http://dx.doi.org/10.2307/2275474
 [E]
Ryszard
Engelking, Dimension theory, NorthHolland Publishing Co.,
AmsterdamOxfordNew York; PWN—Polish Scientific Publishers, Warsaw,
1978. Translated from the Polish and revised by the author; NorthHolland
Mathematical Library, 19. MR 0482697
(58 #2753b)
 [HKL]
L.
A. Harrington, A.
S. Kechris, and A.
Louveau, A GlimmEffros dichotomy for Borel
equivalence relations, J. Amer. Math. Soc.
3 (1990), no. 4,
903–928. MR 1057041
(91h:28023), http://dx.doi.org/10.1090/S08940347199010570415
 [HOR]
R.
Haydon, E.
Odell, and H.
Rosenthal, On certain classes of Baire1 functions with
applications to Banach space theory, Functional analysis (Austin, TX,
1987/1989) Lecture Notes in Math., vol. 1470, Springer, Berlin,
1991, pp. 1–35. MR 1126734
(92h:46018), http://dx.doi.org/10.1007/BFb0090209
 [JM]
Steve
Jackson and R.
Daniel Mauldin, Some complexity results in topology and
analysis, Fund. Math. 141 (1992), no. 1,
75–83. MR
1178370 (93i:03067)
 [JR]
J.
E. Jayne and C.
A. Rogers, First level Borel functions and isomorphisms, J.
Math. Pures Appl. (9) 61 (1982), no. 2,
177–205. MR
673304 (84a:54072)
 [K]
L. Keldi\v{s}, Sur les fonctions premières measurables B, Dokl. Akad. Nauk. SSSR 4 (1934), 192197.
 [KL]
A.
S. Kechris and A.
Louveau, A classification of Baire class 1
functions, Trans. Amer. Math. Soc.
318 (1990), no. 1,
209–236. MR
946424 (90f:26005), http://dx.doi.org/10.1090/S00029947199009464243
 [L1]
Alain
Louveau, A separation theorem for
Σ¹₁ sets, Trans. Amer. Math.
Soc. 260 (1980), no. 2, 363–378. MR 574785
(81j:04001), http://dx.doi.org/10.1090/S0002994719800574785X
 [L2]
A.
S. Kechris, D.
A. Martin, and Y.
N. Moschovakis (eds.), Cabal seminar 79–81, Lecture
Notes in Mathematics, vol. 1019, SpringerVerlag, Berlin, 1983. MR 730583
(86j:03002)
 [MK]
Analytic sets, Academic Press, Inc. [Harcourt Brace Jovanovich,
Publishers], LondonNew York, 1980. Lectures delivered at a Conference held
at University College, University of London, London, July 16–29,
1978. MR
608794 (82m:03063)
 [vMP]
J.
van Mill and R.
Pol, Baire 1 functions which are not countable unions of continuous
functions, Acta Math. Hungar. 66 (1995), no. 4,
289–300. MR 1314008
(96a:54017), http://dx.doi.org/10.1007/BF01876046
 [M]
Michał
Morayne, Algebras of Borel measurable functions, Fund. Math.
141 (1992), no. 3, 229–242. MR 1199236
(93j:26003)
 [R]
Haskell
P. Rosenthal, Some recent discoveries in the
isomorphic theory of Banach spaces, Bull. Amer.
Math. Soc. 84 (1978), no. 5, 803–831. MR 499730
(80d:46023), http://dx.doi.org/10.1090/S000299041978145212
 [R1]
Haskell
Rosenthal, A characterization of Banach spaces
containing 𝑐₀, J. Amer. Math.
Soc. 7 (1994), no. 3, 707–748. MR 1242455
(94i:46032), http://dx.doi.org/10.1090/S08940347199412424554
 [R2]
H.P. Rosenthal, Differences of bounded semicontinuous functions, I (to appear).
 [S]
Sławomir
Solecki, Covering analytic sets by families of closed sets, J.
Symbolic Logic 59 (1994), no. 3, 1022–1031. MR 1295987
(95g:54033), http://dx.doi.org/10.2307/2275926
 [SS]
Saharon
Shelah and Juris
Steprāns, Decomposing Baire class 1 functions into
continuous functions, Fund. Math. 145 (1994),
no. 2, 171–180. MR 1297403
(95i:03110)
 [St]
Juris
Steprāns, A very discontinuous Borel function, J.
Symbolic Logic 58 (1993), no. 4, 1268–1283. MR 1253921
(95c:03120), http://dx.doi.org/10.2307/2275142
 [Sr]
Jacques
Stern, Évaluation du rang de Borel de certains
ensembles, C. R. Acad. Sci. Paris Sér. AB 286
(1978), no. 20, A855–A857 (French, with English summary). MR 498855
(81e:03049)
 [AN]
 S.I. Adyan and P.S. Novikov, On a semicontinuous function, Zap. MPGI W.I. Lenina 138 (1958), 310. MR 22:11081
 [BD]
 H. Becker and R. Dougherty, On disjoint Borel uniformizations, Adv. Math. (to appear).
 [CM]
 J. Cicho\'{n} and M. Morayne, Universal functions and generalized classes of functions, Proc. Amer. Math. Soc. 102 (1988), 8389. MR 89c:26003
 [CMPS]
 J. Cicho\'{n}, M. Morayne, J. Pawlikowski, and S. Solecki, Decomposing Baire functions, J. Symb. Logic 56 (1991), 12731283. MR 92j:04001
 [E]
 R. Engelking, Dimension Theory, NorthHolland, 1978. MR 58:2753b
 [HKL]
 L. Harrington, A.S. Kechris, and A. Louveau, A GlimmEffros dichotomy for Borel equivalence relations, J. Amer. Math. Soc. 3 (1990), 903927. MR 91h:28023
 [HOR]
 R. Haydon, E. Odell, and H. P. Rosenthal, Certain subclasses of Baire1 functions with Banach space applications, Lecture Notes in Mathematics 1470, SpringerVerlag, 1990, pp. 135. MR 92h:46018
 [JM]
 S. Jackson and R.D. Mauldin, Some complexity results in topology and analysis, Fund. Math. 141 (1992), 7583. MR 93i:03067
 [JR]
 J.E. Jayne and C.A. Rogers, First level Borel functions and isomorphisms, J. Math. pures et appl. 61 (1982), 177205. MR 84a:54072
 [K]
 L. Keldi\v{s}, Sur les fonctions premières measurables B, Dokl. Akad. Nauk. SSSR 4 (1934), 192197.
 [KL]
 A.S. Kechris and A. Louveau, A classification of Baire class 1 functions, Trans. Amer. Math. Soc. 318 (1990), 209236. MR 90f:26005
 [L1]
 A. Louveau, A separation theorem for sets, Trans. Amer. Math. Soc. 260 (1980), 363378. MR 81j:04001
 [L2]
 A. Louveau, Some results in the Wadge hierachy of Borel sets, Cabal Seminar 7981, Lecture Notes in Mathematics 1019, SpringerVerlag, 1983, pp. 2855. MR 86j:03002
 [MK]
 D.A. Martin and A.S. Kechris, Infinite games and effective descriptive set theory, Analytic Sets, Academic Press, London, 1980, pp. 403470. MR 82m:03063
 [vMP]
 J. van Mill and R. Pol, Baire 1 functions which are not countable unions of continuous functions, Acta Math. Hungar. 66 (1995), 289300. MR 96a:54017
 [M]
 M. Morayne, Algebras of Borel measurable functions, Fund. Math. 141 (1992), 229242. MR 93j:26003
 [R]
 H.P. Rosenthal, Some recent discoveries in the isomorphic theory of Banach spaces, Bull. Amer. Math. Soc. 84 (1978), 803831. MR 80d:46023
 [R1]
 H.P. Rosenthal, A characterization of Banach spaces containing , J. Amer. Math. Soc. 7 (1994), 707748. MR 94i:46032
 [R2]
 H.P. Rosenthal, Differences of bounded semicontinuous functions, I (to appear).
 [S]
 S. Solecki, Covering analytic sets by families of closed sets, J. Symb. Logic 59 (1994), 10221031. MR 95g:54033
 [SS]
 S. Shelah and J. Stepr\={a}ns, Decomposing Baire class 1 functions into continuous functions, Fund. Math. 145 (1994), 171180. MR 95i:03110
 [St]
 J. Stepr\={a}ns, A very discontinuous Borel function, J. Symb. Logic 58 (1993), 12681283. MR 95c:03120
 [Sr]
 J. Stern, Évaluation du rang de Borel de certains ensembles, C. R. Acad. Sc. Paris Série A 286 (1978), 855857. MR 81e:03049
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Additional Information
Slawomir Solecki
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email:
ssolecki@indiana.edu
DOI:
http://dx.doi.org/10.1090/S0894034798002690
PII:
S 08940347(98)002690
Keywords:
Baire class 1 functions,
Borel sets,
semicontinuous functions,
Borel measures,
covering of the meager ideal,
decomposition of functions
Received by editor(s):
May 1, 1997
Article copyright:
© Copyright 1998
American Mathematical Society
