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Decomposing Borel sets and functions
and the structure of Baire Class 1 functions

Author: Slawomir Solecki
Journal: J. Amer. Math. Soc. 11 (1998), 521-550
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 $A$ is either $\boldsymbol \Sigma ^{0}_{\alpha }$ or there is a continuous injection $\phi :\; \omega ^{\omega }\to A$ such that for any $\boldsymbol \Sigma ^{0}_{\alpha }$ set $B\subset A$, $\phi ^{-1}(B)$ 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.

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Additional Information

Slawomir Solecki
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405

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

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