# Essential range

In mathematics, particularly measure theory, the essential range, or the set of essential values, of a function is intuitively the 'non-negligible' range of the function: It does not change between two functions that are equal almost everywhere. One way of thinking of the essential range of a function is the set on which the range of the function is 'concentrated'.

## Formal definition

Let $(X,{\cal {A}},\mu )$ be a measure space, and let $(Y,{\cal {T}})$ be a topological space. For any $({\cal {A}},\sigma ({\cal {T}}))$-measurable function $f:X\to Y$, we say the essential range of $f$ to mean the set
    $\operatorname {ess.im} (f)=\left\{y\in Y\mid 0<\mu (f^{-1}(U)){\text{ for all }}U\in {\cal {T}}{\text{ with }}y\in U\right\}.$$^{:Example0.A.5}$
Equivalently, $\operatorname {ess.im} (f)=\operatorname {supp} (f_{*}\mu )$, where $f_{*}\mu$ is the pushforward measure onto $\sigma ({\cal {T}})$ of $\mu$ under $f$ and $\operatorname {supp} (f_{*}\mu )$ denotes the support of $f_{*}\mu .$

### Essential values

The phrase "essential value of $f$" is sometimes used to mean an element of the essential range of $f.$$^{:Exercise4.1.6}$$^{:Example7.1.11}$

## Special cases of common interest
### Y = C

Say $(Y,{\cal {T}})$ is $\mathbb {C}$ equipped with its usual topology. Then the essential range of f is given by
    $\operatorname {ess.im} (f)=\left\{z\in \mathbb {C} \mid {\text{for all}}\ \varepsilon \in \mathbb {R} _{>0}:0<\mu \{x\in X:|f(x)-z|<\varepsilon \}\right\}.$$^{:Definition4.36}$$^{:cf.Exercise6.11}$
In other words: The essential range of a complex-valued function is the set of all complex numbers z such that the inverse image of each ε-neighbourhood of z under f has positive measure.

### (Y,T) is discrete

Say $(Y,{\cal {T}})$ is discrete, i.e., ${\cal {T}}={\cal {P}}(Y)$ is the power set of $Y,$ i.e., the discrete topology on $Y.$ Then the essential range of f is the set of values y in Y with strictly positive $f_{*}\mu$-measure:
    $\operatorname {ess.im} (f)=\{y\in Y:0<\mu (f^{\text{pre}}\{y\})\}=\{y\in Y:0<(f_{*}\mu )\{y\}\}.$$^{:Example1.1.29}$

## Properties

* The essential range of a measurable function, being the support of a measure, is always closed.
* The essential range ess.im(f) of a measurable function is always a subset of ${\overline {\operatorname {im} (f)}}$.
* The essential image cannot be used to distinguish functions that are almost everywhere equal: If $f=g$ holds $\mu$-almost everywhere, then $\operatorname {ess.im} (f)=\operatorname {ess.im} (g)$.
* These two facts characterise the essential image: It is the biggest set contained in the closures of $\operatorname {im} (g)$ for all g that are a.e. equal to f:
       $\operatorname {ess.im} (f)=\bigcap _{f=g\,{\text{a.e.}}}{\overline {\operatorname {im} (g)}}$.
* The essential range satisfies $\forall A\subseteq X:f(A)\cap \operatorname {ess.im} (f)=\emptyset \implies \mu (A)=0$.
* This fact characterises the essential image: It is the smallest closed subset of $\mathbb {C}$ with this property.
* The essential supremum of a real valued function equals the supremum of its essential image and the essential infimum equals the infimum of its essential range. Consequently, a function is essentially bounded if and only if its essential range is bounded.
* The essential range of an essentially bounded function f is equal to the spectrum $\sigma (f)$ where f is considered as an element of the C*-algebra $L^{\infty }(\mu )$.

## Examples

* If $\mu$ is the zero measure, then the essential image of all measurable functions is empty.
* This also illustrates that even though the essential range of a function is a subset of the closure of the range of that function, equality of the two sets need not hold.
* If $X\subseteq \mathbb {R} ^{n}$ is open, $f:X\to \mathbb {C}$ continuous and $\mu$ the Lebesgue measure, then $\operatorname {ess.im} (f)={\overline {\operatorname {im} (f)}}$ holds. This holds more generally for all Borel measures that assign non-zero measure to every non-empty open set.

## Extension

The notion of essential range can be extended to the case of $f:X\to Y$, where $Y$ is a separable metric space. If $X$ and $Y$ are differentiable manifolds of the same dimension, if $f\in$ VMO$(X,Y)$ and if $\operatorname {ess.im} (f)\neq Y$, then $\deg f=0$.