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 | References | Similar Articles | Additional Information

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.

- 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