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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 $Z$?


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up vote 3 down vote accepted


1) If $s \in S$, $\varepsilon > 0$ and $u = f$ a.e., there are lots of points $x$ with $|u(x)-s| < \varepsilon$.

2) Conversely, if $t \notin S$, define a suitable function $u$ that differs from $f$ only when $f(x)$ is near $t$.

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Very thanks for help. – Richard Dec 20 '11 at 8:34

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