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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Transformations into Baire $ 1$ functions


Authors: A. M. Bruckner, Roy O. Davies and C. Goffman
Journal: Proc. Amer. Math. Soc. 67 (1977), 62-66
MSC: Primary 26A21
DOI: https://doi.org/10.1090/S0002-9939-1977-0480903-X
MathSciNet review: 0480903
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Abstract: A measurable f from $ I = [0,1]$ to R is equivalent to a Baire 2 function but may not be equivalent to any Baire 1 function. Gorman has obtained the following interesting contrasting facts. If f assumes finitely many values there is a homeomorphism h of I such that $ f \circ h$ is equivalent to a Baire 1 function, but there is a measurable f which assumes countably many values which does not have this property. However, the example of Gorman is such that for some homeomorphisms h the function $ f \circ h$ is not measurable. It is shown here that if $ f \circ h$ is measurable, for every homeomorphism h, then there is an h for which $ f \circ h$ is equivalent to a Baire 1 function.


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DOI: https://doi.org/10.1090/S0002-9939-1977-0480903-X
Article copyright: © Copyright 1977 American Mathematical Society