# Essential range

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Concept in measure theory

In [mathematics](/source/Mathematics), particularly [measure theory](/source/Measure_theory), the **essential range**, or the set of **essential values**, of a [function](/source/Function_(mathematics)) is intuitively the 'non-negligible' [range of the function](/source/Range_of_a_function): It does not change between two functions that are equal [almost everywhere](/source/Almost_everywhere). One way of thinking of the essential range of a function is the [set](/source/Set_(mathematics)) on which the range of the function is 'concentrated'.

## Formal definition

Let ( X , A , μ ) {\displaystyle (X,{\cal {A}},\mu )} be a [measure space](/source/Measure_space), and let ( Y , T ) {\displaystyle (Y,{\cal {T}})} be a [topological space](/source/Topological_space). For any ( A , σ ( T ) ) {\displaystyle ({\cal {A}},\sigma ({\cal {T}}))} -[measurable function](/source/Measurable_function) f : X → Y {\displaystyle f:X\to Y} , we say the **essential range** of f {\displaystyle f} to mean the set

- e s s . i m ⁡ ( f ) = { y ∈ Y ∣ 0 < μ ( f − 1 ( U ) ) for all U ∈ T with y ∈ U } . {\displaystyle \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\}.} [1]: Example 0.A.5[2][3]

Equivalently, e s s . i m ⁡ ( f ) = supp ⁡ ( f ∗ μ ) {\displaystyle \operatorname {ess.im} (f)=\operatorname {supp} (f_{*}\mu )} , where f ∗ μ {\displaystyle f_{*}\mu } is the [pushforward measure](/source/Pushforward_measure) onto σ ( T ) {\displaystyle \sigma ({\cal {T}})} of μ {\displaystyle \mu } under f {\displaystyle f} and supp ⁡ ( f ∗ μ ) {\displaystyle \operatorname {supp} (f_{*}\mu )} denotes the [support](/source/Support_(measure_theory)) of f ∗ μ . {\displaystyle f_{*}\mu .} [4]

### Essential values

The phrase "**essential value** of f {\displaystyle f} " is sometimes used to mean an element of the essential range of f . {\displaystyle f.} [5]: Exercise 4.1.6[6]: Example 7.1.11

## Special cases of common interest

### *Y* = **C**

Say ( Y , T ) {\displaystyle (Y,{\cal {T}})} is C {\displaystyle \mathbb {C} } equipped with its usual topology. Then the essential range of *f* is given by

- e s s . i m ⁡ ( f ) = { z ∈ C ∣ for all ε ∈ R > 0 : 0 < μ { x ∈ X : | f ( x ) − z | < ε } } . {\displaystyle \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\}.} [7]: Definition 4.36[8][9]: cf. Exercise 6.11[10]: Exercise 3.19[11]: Definition 2.61

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 , T ) {\displaystyle (Y,{\cal {T}})} is [discrete](/source/Discrete_space), i.e., T = P ( Y ) {\displaystyle {\cal {T}}={\cal {P}}(Y)} is the [power set](/source/Power_set) of Y , {\displaystyle Y,} i.e., the [discrete topology](/source/Discrete_space#Definition) on Y . {\displaystyle Y.} Then the essential range of *f* is the set of values *y* in *Y* with strictly positive f ∗ μ {\displaystyle f_{*}\mu } -measure:

- e s s . i m ⁡ ( f ) = { y ∈ Y : 0 < μ ( f pre { y } ) } = { y ∈ Y : 0 < ( f ∗ μ ) { y } } . {\displaystyle \operatorname {ess.im} (f)=\{y\in Y:0<\mu (f^{\text{pre}}\{y\})\}=\{y\in Y:0<(f_{*}\mu )\{y\}\}.} [12]: Example 1.1.29[13][14]

## Properties

- The essential range of a measurable function, being the [support of a measure](/source/Support_(measure_theory)), is always closed.

- The essential range ess.im(f) of a measurable function is always a subset of im ⁡ ( f ) ¯ {\displaystyle {\overline {\operatorname {im} (f)}}} .

- The essential image cannot be used to distinguish functions that are almost everywhere equal: If f = g {\displaystyle f=g} holds μ {\displaystyle \mu } -[almost everywhere](/source/Almost_everywhere), then e s s . i m ⁡ ( f ) = e s s . i m ⁡ ( g ) {\displaystyle \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 im ⁡ ( g ) {\displaystyle \operatorname {im} (g)} for all g that are a.e. equal to f:

- - e s s . i m ⁡ ( f ) = ⋂ f = g a.e. im ⁡ ( g ) ¯ {\displaystyle \operatorname {ess.im} (f)=\bigcap _{f=g\,{\text{a.e.}}}{\overline {\operatorname {im} (g)}}} .

- The essential range satisfies ∀ A ⊆ X : f ( A ) ∩ e s s . i m ⁡ ( f ) = ∅ ⟹ μ ( A ) = 0 {\displaystyle \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 C {\displaystyle \mathbb {C} } with this property.

- The [essential supremum](/source/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](/source/Spectrum_(functional_analysis)#Spectrum_of_a_unital_Banach_algebra) σ ( f ) {\displaystyle \sigma (f)} where f is considered as an element of the [C*-algebra](/source/C*-algebra) L ∞ ( μ ) {\displaystyle L^{\infty }(\mu )} .

## Examples

- If μ {\displaystyle \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 ⊆ R n {\displaystyle X\subseteq \mathbb {R} ^{n}} is open, f : X → C {\displaystyle f:X\to \mathbb {C} } continuous and μ {\displaystyle \mu } the [Lebesgue measure](/source/Lebesgue_measure), then e s s . i m ⁡ ( f ) = im ⁡ ( f ) ¯ {\displaystyle \operatorname {ess.im} (f)={\overline {\operatorname {im} (f)}}} holds. This holds more generally for all [Borel measures](/source/Borel_measure) 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 → Y {\displaystyle f:X\to Y} , where Y {\displaystyle Y} is a [separable](/source/Separable_space) [metric space](/source/Metric_space). If X {\displaystyle X} and Y {\displaystyle Y} are [differentiable manifolds](/source/Differentiable_manifold) of the same dimension, if f ∈ {\displaystyle f\in } [VMO](/source/Bounded_mean_oscillation#The_space_VMO) ( X , Y ) {\displaystyle (X,Y)} and if e s s . i m ⁡ ( f ) ≠ Y {\displaystyle \operatorname {ess.im} (f)\neq Y} , then deg ⁡ f = 0 {\displaystyle \deg f=0} .[15]

## See also

- [Essential supremum and essential infimum](/source/Essential_supremum_and_essential_infimum)

- [measure](/source/Measure_(mathematics))

- [Lp space](/source/Lp_space)

## References

1. **[^](#cite_ref-1)** [Zimmer, Robert J.](/source/Robert_Zimmer) (1990). *Essential Results of Functional Analysis*. University of Chicago Press. p. 2. [ISBN](/source/ISBN_(identifier)) [0-226-98337-4](https://en.wikipedia.org/wiki/Special:BookSources/0-226-98337-4).

1. **[^](#cite_ref-2)** [Kuksin, Sergei](/source/Sergei_B._Kuksin); Shirikyan, Armen (2012). *Mathematics of Two-Dimensional Turbulence*. Cambridge University Press. p. 292. [ISBN](/source/ISBN_(identifier)) [978-1-107-02282-9](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-02282-9).

1. **[^](#cite_ref-3)** Kon, Mark A. (1985). *Probability Distributions in Quantum Statistical Mechanics*. Springer. pp. 74, 84. [ISBN](/source/ISBN_(identifier)) [3-540-15690-9](https://en.wikipedia.org/wiki/Special:BookSources/3-540-15690-9).

1. **[^](#cite_ref-4)** Driver, Bruce (May 7, 2012). [*Analysis Tools with Examples*](https://mathweb.ucsd.edu/~bdriver/240C-S2018/Lecture_Notes/2012%20Notes/240Lecture_Notes_Ver8.pdf) (PDF). p. 327. Cf. Exercise 30.5.1.

1. **[^](#cite_ref-5)** [Segal, Irving E.](/source/Irving_Segal); [Kunze, Ray A.](/source/Ray_Kunze) (1978). *Integrals and Operators* (2nd revised and enlarged ed.). Springer. p. 106. [ISBN](/source/ISBN_(identifier)) [0-387-08323-5](https://en.wikipedia.org/wiki/Special:BookSources/0-387-08323-5).

1. **[^](#cite_ref-6)** Bogachev, Vladimir I.; Smolyanov, Oleg G. (2020). *Real and Functional Analysis*. Moscow Lectures. Springer. p. 283. [ISBN](/source/ISBN_(identifier)) [978-3-030-38219-3](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-38219-3). [ISSN](/source/ISSN_(identifier)) [2522-0314](https://search.worldcat.org/issn/2522-0314).

1. **[^](#cite_ref-7)** Weaver, Nik (2013). *Measure Theory and Functional Analysis*. World Scientific. p. 142. [ISBN](/source/ISBN_(identifier)) [978-981-4508-56-8](https://en.wikipedia.org/wiki/Special:BookSources/978-981-4508-56-8).

1. **[^](#cite_ref-8)** [Bhatia, Rajendra](/source/Rajendra_Bhatia) (2009). *Notes on Functional Analysis*. Hindustan Book Agency. p. 149. [ISBN](/source/ISBN_(identifier)) [978-81-85931-89-0](https://en.wikipedia.org/wiki/Special:BookSources/978-81-85931-89-0).

1. **[^](#cite_ref-9)** [Folland, Gerald B.](/source/Gerald_Folland) (1999). *Real Analysis: Modern Techniques and Their Applications*. Wiley. p. 187. [ISBN](/source/ISBN_(identifier)) [0-471-31716-0](https://en.wikipedia.org/wiki/Special:BookSources/0-471-31716-0).

1. **[^](#cite_ref-10)** Rudin, Walter (1987). *Real and complex analysis* (3rd ed.). New York: McGraw-Hill. [ISBN](/source/ISBN_(identifier)) [0-07-054234-1](https://en.wikipedia.org/wiki/Special:BookSources/0-07-054234-1).

1. **[^](#cite_ref-11)** Douglas, Ronald G. (1998). *Banach algebra techniques in operator theory* (2nd ed.). New York Berlin Heidelberg: Springer. [ISBN](/source/ISBN_(identifier)) [0-387-98377-5](https://en.wikipedia.org/wiki/Special:BookSources/0-387-98377-5).

1. **[^](#cite_ref-12)** Cf. [Tao, Terence](/source/Terence_Tao) (2012). *Topics in Random Matrix Theory*. American Mathematical Society. p. 29. [ISBN](/source/ISBN_(identifier)) [978-0-8218-7430-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8218-7430-1).

1. **[^](#cite_ref-13)** Cf. [Freedman, David](/source/David_A._Freedman) (1971). *Markov Chains*. Holden-Day. p. 1.

1. **[^](#cite_ref-14)** Cf. [Chung, Kai Lai](/source/Chung_Kai-lai) (1967). *Markov Chains with Stationary Transition Probabilities*. Springer. p. 135.

1. **[^](#cite_ref-15)** [Brezis, Haïm](/source/Ha%C3%AFm_Brezis); [Nirenberg, Louis](/source/Louis_Nirenberg) (September 1995). "Degree theory and BMO. Part I: Compact manifolds without boundaries". *Selecta Mathematica*. **1** (2): 197–263. [doi](/source/Doi_(identifier)):[10.1007/BF01671566](https://doi.org/10.1007%2FBF01671566).

- [Walter Rudin](/source/Walter_Rudin) (1974). [*Real and Complex Analysis*](https://archive.org/details/realcomplexanaly00rudi_0) (2nd ed.). [McGraw-Hill](/source/McGraw-Hill). [ISBN](/source/ISBN_(identifier)) [978-0-07-054234-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-054234-1).

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