Let $\mu$ be a nonnegative measure on a $\sigma$-algebra of subsets of $X$ and let $f \colon X\to \mathbb{C}$ be in $L^\infty$.
The essential range $S$ of $f$ is defined ( Wikipedia) as a set of all $z \in \mathbb{C}$ such that for all $\varepsilon>0$: $$\mu ( \{x\in X: |f(x)-z| < \varepsilon \})>0.$$
Let's consider the following set $$Z:=\bigcap \overline{u(X)},$$ where intersection is taken over all measurable functions $u\colon X\to \mathbb{C}$ such that $u(x)=f(x)$ almost everywhere.
Is there a connection beetween sets $S$ and $Z$?
Thanks.