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Formal definition

Let ( X , A , μ ) {\displaystyle (X,{\cal {A}},\mu )} be a measure space, and let ( Y , T ) {\displaystyle (Y,{\cal {T}})} be a topological space. For any ( A , σ ( T ) ) {\displaystyle ({\cal {A}},\sigma ({\cal {T}}))} -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 ²³

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 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 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 ⁸⁹: cf. Exercise 6.11 

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 , T ) {\displaystyle (Y,{\cal {T}})} is discrete, i.e., T = P ( Y ) {\displaystyle {\cal {T}}={\cal {P}}(Y)} is the power set of Y , {\displaystyle Y,} i.e., the discrete topology 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\}\}.} ¹⁰: Example 1.1.29 ¹¹¹²

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 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, 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 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 σ ( f ) {\displaystyle \sigma (f)} where f is considered as an element of the 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, 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 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 metric space. If X {\displaystyle X} and Y {\displaystyle Y} are differentiable manifolds of the same dimension, if f ∈ {\displaystyle f\in } 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} .¹³

See also

• Essential supremum and essential infimum
• measure
• Lp space

• Walter Rudin (1974). Real and Complex Analysis (2nd ed.). McGraw-Hill. ISBN 978-0-07-054234-1.

References

1. Zimmer, Robert J. (1990). Essential Results of Functional Analysis. University of Chicago Press. p. 2. ISBN 0-226-98337-4. 0-226-98337-4 ↩

2. Kuksin, Sergei; Shirikyan, Armen (2012). Mathematics of Two-Dimensional Turbulence. Cambridge University Press. p. 292. ISBN 978-1-107-02282-9. 978-1-107-02282-9 ↩

3. Kon, Mark A. (1985). Probability Distributions in Quantum Statistical Mechanics. Springer. pp. 74, 84. ISBN 3-540-15690-9. 3-540-15690-9 ↩

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

5. Segal, Irving E.; Kunze, Ray A. (1978). Integrals and Operators (2nd revised and enlarged ed.). Springer. p. 106. ISBN 0-387-08323-5. 0-387-08323-5 ↩

6. Bogachev, Vladimir I.; Smolyanov, Oleg G. (2020). Real and Functional Analysis. Moscow Lectures. Springer. p. 283. ISBN 978-3-030-38219-3. ISSN 2522-0314. 978-3-030-38219-3 ↩

7. Weaver, Nik (2013). Measure Theory and Functional Analysis. World Scientific. p. 142. ISBN 978-981-4508-56-8. 978-981-4508-56-8 ↩

8. Bhatia, Rajendra (2009). Notes on Functional Analysis. Hindustan Book Agency. p. 149. ISBN 978-81-85931-89-0. 978-81-85931-89-0 ↩

9. Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications. Wiley. p. 187. ISBN 0-471-31716-0. 0-471-31716-0 ↩

10. Cf. Tao, Terence (2012). Topics in Random Matrix Theory. American Mathematical Society. p. 29. ISBN 978-0-8218-7430-1. 978-0-8218-7430-1 ↩

11. Cf. Freedman, David (1971). Markov Chains. Holden-Day. p. 1. /wiki/David_A._Freedman ↩

12. Cf. Chung, Kai Lai (1967). Markov Chains with Stationary Transition Probabilities. Springer. p. 135. /wiki/Chung_Kai-lai ↩

13. Brezis, Haïm; Nirenberg, Louis (September 1995). "Degree theory and BMO. Part I: Compact manifolds without boundaries". Selecta Mathematica. 1 (2): 197–263. doi:10.1007/BF01671566. /wiki/Doi_(identifier) ↩