On bounded functions satisfying averaging conditions. II

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.