Let $A_f$ be the set of all averages $\frac{1}{\mu(E)}\intop_{E}\,f\,d\mu$
where $E$ is of positive measure. What is the relationship between $A_f$ and $\mathbb{R}_f$? Is $A_f$ always closed?
Are there measures $\mu$ such that $A_f$ is convex for every $f\in L^\infty (\mu)$? Are there measure $\mu$ such that $A_f$ fails to convex for some $f\in L^\infty (\mu)$?
My guess is that $A_f\subset R_f$ and it is closed. But I am not sure if it is convex???
Answers.
a. The essential range is not necessarily achieved by averages. For examples if $f(x)=x$, $x\in [0,1]$, then the values 0 and 1 are not achieved by $A_f$.
b. Indeed the range of $A_f$ is not necessarily convex. Let $\mu=\delta_0+\delta_1$, then the only values it achieves are $f(0)$, $f(1)$ and $\frac{1}{2}\big(f(0)+f(1)\big)$.
It is convex in the case of Lebesgue measure on an open subset of a Euclidean space.