Classes of Baire functions

Cox, Gregory V.; Humke, Paul D.

doi:10.1090/S0002-9947-1982-0637714-8

Classes of Baire functions
HTML articles powered by AMS MathViewer

by Gregory V. Cox and Paul D. Humke PDF

Trans. Amer. Math. Soc. 269 (1982), 627-635 Request permission

Abstract:

Let $\mathcal {A}$ and $\mathcal {P}$ denote the sets of approximately continuous and almost everywhere continuous functions, and ${B_1}(F)$ denote Baire’s first class generated by $F$. The classes ${B_1}(\mathcal {A})$, ${B_1}(\mathcal {P})$, ${B_1}(\mathcal {A} \cap \mathcal {P})$, and Grande’s class $\mathcal {A}{\mathcal {P}_1}$ are investigated in some detail. Although Grande’s question of whether ${B_1}(\mathcal {A} \cap \mathcal {P}) = {B_1}(\mathcal {A}) \cap {B_1}(\mathcal {A}) \cap \mathcal {A}{\mathcal {P}_1}$ is not settled, we do show, among other results, that $\mathcal {A}{\mathcal {P}_1} \subset {B_1}(\mathcal {P})$.

References

G. V. Cox and P. D. Humke, A note on pointwise limits of functions which are approximately continuous and almost everywhere continuous, Studia Sci. Math. Hungar. 14 (1979), no. 4, 319–323. MR 685898
Casper Goffman and Daniel Waterman, Approximately continuous transformations, Proc. Amer. Math. Soc. 12 (1961), 116–121. MR 120327, DOI 10.1090/S0002-9939-1961-0120327-6
Zbigniew Grande, Sur les suites de fonctions approximativement continues et continues presque partout, Colloq. Math. 38 (1977/78), no. 2, 259–262 (French). MR 580932, DOI 10.4064/cm-38-2-259-262
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
R. Daniel Mauldin, $\sigma$-ideals and related Baire systems, Fund. Math. 71 (1971), no. 2, 171–177. MR 293027, DOI 10.4064/fm-71-2-171-177

The topology of almost everywhere continuous, approximately continuous functions

Richard J. O’Malley, Approximately differentiable functions: the $r$ topology, Pacific J. Math. 72 (1977), no. 1, 207–222. MR 447499
R. J. O’Malley, Approximately continuous functions which are continuous almost everywhere, Acta Math. Acad. Sci. Hungar. 33 (1979), no. 3-4, 395–402. MR 542489, DOI 10.1007/BF01902575
David Preiss, Limits of approximately continuous functions, Czechoslovak Math. J. 21(96) (1971), 371–372. MR 286947