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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
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 \[ \lim \limits _{r \to 0} \frac {1}{{|B(x,r)|}} {\int } _{B(x,r)}f(y)dy = f(x)\] 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?)

  • Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • Zygmunt Zahorski, Über die Menge der Punkte in welchen die Ableitung unendlich ist, Tôhoku Math. J. 48 (1941), 321–330 (German). MR 27825

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Keywords: Approximately continuous, extreme points, <!– MATH ${G_\delta }$ –> <IMG WIDTH="30" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${G_\delta }$"> sets of measure 0 in <IMG WIDTH="32" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="${R^n}$">
Article copyright: © Copyright 1976 American Mathematical Society