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Approximate Continuity and Differentiation

Published online by Cambridge University Press:  20 November 2018

Maurice Sion
Maurice Sion*
Affiliation:
University of British Columbia
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The relation between the notions of measurability and continuity of a function has received a great deal of attention. The best-known result in this connection is the Vitali-Lusin theorem. Various versions of it can be found in (2; 3; 5; 8; 9). We prove one in this paper (Theorem 3.5) under very weak assumptions and state the classical one in Corollary 3.6. Functions satisfying the property stated in the theorem are frequently called quasicontinuous.

Here we are interested in the notion of approximate continuity, which we call μ-continuity for short and which is closely connected to differentiation. It was first introduced by Lebesgue (4) and it has so far required knowledge of density theorems in order to prove its relation to measurability. In this paper, making use of a certain type of Vitali property, we prove first the relation between μ-measurable functions and those μ-continuous almost everywhere (Theorems 3.8 and 3.9). Then, in §4, with Theorem 3.8 as the main tool, we derive several theorems about density and differentiation which extend results found in (1; 3; 7; 8).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 467 - 475
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Besicovitch, A. S., On the fundamental geometrical properties of linearly measurable plane sets of points (II), Math. Ann., 115 (1938), 296–329.Google Scholar
2. Bertolini, F., Il problema di Lusin, Ricerche Mat., 6 (1957), 288–306.Google Scholar
3. Hahn, H. and Rosenthal, A., Set functions (University of New Mexico Press, 1948).Google Scholar
4. Lebesgue, H., Sur une propriété des fonctions, C. R. Acad. Sci. Paris, 157 (1903), 1228.Google Scholar
5. Letta, G., Il problema di Vitali-Lusin negli spazi perfettamente normali, Ricerche Mat., 8 (1959), 128–137.Google Scholar
6. Mickle, E. J. and Rado, T., Density theorems for outer measures in n-space, Proc. Amer. Math. Soc, 9 (1958), 433–439.Google Scholar
7. Morse, A. P. and Randolph, J. F., The p rectifiable subsets of the plane, Trans. Amer. Math. Soc, 55 (1944), 236–305.Google Scholar
8. Saks, S., Theory of the integral (Warsaw, 1937).Google Scholar
9. Schaerf, H. M., On the continuity of measurable functions in neighborhood spaces, Portugaliae Math., 6 (1947), 33–44 and 66; 7 (1948), 91-92.Google Scholar