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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On bounded functions satisfying averaging conditions. II


Author: Rotraut Goubau Cahill
Journal: Trans. Amer. Math. Soc. 223 (1976), 295-304
MSC: Primary 26A69; Secondary 31B05
DOI: https://doi.org/10.1090/S0002-9947-1976-0422539-6
MathSciNet review: 0422539
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Abstract: Let $ S(f)$ denote the subspace of $ {L^\infty }({R^n})$ consisting of those real valued functions f for which

$\displaystyle \mathop {\lim }\limits_{r \to 0} \frac{1}{{\vert B(x,r)\vert}} {\int} _{B(x,r)}f(y)dy = f(x)$

for all x in $ {R^n}$ and let $ L(f)$ be the subspace of $ S(f)$ consisting of the approximately continuous functions. A number of results concerning the existence of functions in $ S(f)$ and $ L(f)$ with special properties are obtained. The extreme points of the unit balls of both spaces are characterized and it is shown that $ L(f)$ is not a dual space. As a preliminary step, it is shown that if E is a $ {G_\delta }$ set of measure 0 in $ {R^n}$, then the complement of E can be decomposed into a collection of closed sets in a particularly useful way.

References [Enhancements On Off] (What's this?)

  • [1] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [2] Zygmunt Zahorski, Über die Menge der Punkte in welchen die Ableitung unendlich ist, Tôhoku Math. J. 48 (1941), 321–330 (German). MR 0027825

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0422539-6
Keywords: Approximately continuous, extreme points, $ {G_\delta }$ sets of measure 0 in $ {R^n}$
Article copyright: © Copyright 1976 American Mathematical Society

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