The Baire order of the functions continuous almost everywhere

Author:
R. Daniel Mauldin

Journal:
Proc. Amer. Math. Soc. **51** (1975), 371-377

MSC:
Primary 26A21

DOI:
https://doi.org/10.1090/S0002-9939-1975-0372128-1

MathSciNet review:
0372128

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Abstract: Let be a complete and separable metric space and a -finite, complete Borel measure on with . Let be the family of all real-valued functions defined on whose set of points of discontinuity is of -measure 0. Let be the functions of Baire's class generated by . It is shown that if and only if is a purely atomic measure whose set of atoms forms a scattered subset of and that if , then the Baire order of is ; in other words, if , then . This answers a generalized version of a problem raised by Sierpinski and Felsztyn. An example is given of a normal space with Borel order 2 and Baire order .

**[1]**K. Kuratowski,*Topology*. Vol. I, PWN, Warsaw; Academic Press, New York, 1966. MR**36**#840. MR**0217751 (36:840)****[2]**R. D. Mauldin,*-ideals and related Baire systems*, Fund. Math.**71**(1971), 171-177. MR**45**#2107. MR**0293027 (45:2107)****[3]**R. D. Mauldin,*Some examples of -ideals and related Baire systems*, Fund. Math.**71**(1971), 179-184. MR**45**#2108. MR**0293028 (45:2108)****[4]**-,*The Baire order of the functions continuous almost everywhere*, Proc. Amer. Math. Soc.**41**(1973), 535-540. MR**48**#2319. MR**0323966 (48:2319)****[5]**W. Sierpinski and T. Felsztyn,*Probleme***10**, Fund. Math.**1**(1920), 224.**[6]**E. Michael,*The product of a normal space and a metric space need not be normal*, Bull. Amer. Math. Soc.**69**(1963), pp. 375-376. MR**27**#2956. MR**0152985 (27:2956)****[7]**S. Willard,*Some examples in the theory of Borel sets*, Fund. Math.**71**(1971), 187-191. MR**0295279 (45:4347)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1975-0372128-1

Keywords:
Borel measure,
Baire's class ,
0-dimension,
scattered

Article copyright:
© Copyright 1975
American Mathematical Society