Essential range

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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 [math]\displaystyle{ (X,{\cal A},\mu) }[/math] be a measure space, and let [math]\displaystyle{ (Y,{\cal T}) }[/math] be a topological space. For any [math]\displaystyle{ ({\cal A},\sigma({\cal T})) }[/math]-measurable [math]\displaystyle{ f:X\to Y }[/math], we say the essential range of [math]\displaystyle{ f }[/math] to mean the set

[math]\displaystyle{ \operatorname{ess.im}(f) = \left\{y\in Y\mid0\lt \mu(f^{-1}(U))\text{ for all }U\in{\cal T} \text{ with } y \in U\right\}. }[/math][1](Example 0.A.5)[2][3]

Equivalently, [math]\displaystyle{ \operatorname{ess.im}(f)=\operatorname{supp}(f_*\mu) }[/math], where [math]\displaystyle{ f_*\mu }[/math] is the pushforward measure onto [math]\displaystyle{ \sigma({\cal T}) }[/math] of [math]\displaystyle{ \mu }[/math] under [math]\displaystyle{ f }[/math] and [math]\displaystyle{ \operatorname{supp}(f_*\mu) }[/math] denotes the support of [math]\displaystyle{ f_*\mu. }[/math][4]

Essential values

We sometimes use the phrase "essential value of [math]\displaystyle{ f }[/math]" to mean an element of the essential range of [math]\displaystyle{ f. }[/math][5](Exercise 4.1.6)[6](Example 7.1.11)

Special cases of common interest

Y = C

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

[math]\displaystyle{ \operatorname{ess.im}(f) = \left\{z \in \mathbb{C} \mid \text{for all}\ \varepsilon\in\mathbb R_{\gt 0}: 0\lt \mu\{x\in X: |f(x) - z| \lt \varepsilon\}\right\}. }[/math][7](Definition 4.36)[8][9](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 [math]\displaystyle{ (Y,{\cal T}) }[/math] is discrete, i.e., [math]\displaystyle{ {\cal T}={\cal P}(Y) }[/math] is the power set of [math]\displaystyle{ Y, }[/math] i.e., the discrete topology on [math]\displaystyle{ Y. }[/math] Then the essential range of f is the set of values y in Y with strictly positive [math]\displaystyle{ f_*\mu }[/math]-measure:

[math]\displaystyle{ \operatorname{ess.im}(f)=\{y\in Y:0\lt \mu(f^\text{pre}\{y\})\}=\{y\in Y:0\lt (f_*\mu)\{y\}\}. }[/math][10](Example 1.1.29)[11][12]

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 [math]\displaystyle{ \overline{\operatorname{im}(f)} }[/math].
  • The essential image cannot be used to distinguish functions that are almost everywhere equal: If [math]\displaystyle{ f=g }[/math] holds [math]\displaystyle{ \mu }[/math]-almost everywhere, then [math]\displaystyle{ \operatorname{ess.im}(f)=\operatorname{ess.im}(g) }[/math].
  • These two facts characterise the essential image: It is the biggest set contained in the closures of [math]\displaystyle{ \operatorname{im}(g) }[/math] for all g that are a.e. equal to f:
[math]\displaystyle{ \operatorname{ess.im}(f) = \bigcap_{f=g\,\text{a.e.}} \overline{\operatorname{im}(g)} }[/math].
  • The essential range satisfies [math]\displaystyle{ \forall A\subseteq X: f(A) \cap \operatorname{ess.im}(f) = \emptyset \implies \mu(A) = 0 }[/math].
  • This fact characterises the essential image: It is the smallest closed subset of [math]\displaystyle{ \mathbb{C} }[/math] 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 [math]\displaystyle{ \sigma(f) }[/math] where f is considered as an element of the C*-algebra [math]\displaystyle{ L^\infty(\mu) }[/math].

Examples

  • If [math]\displaystyle{ \mu }[/math] 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 [math]\displaystyle{ X\subseteq\mathbb{R}^n }[/math] is open, [math]\displaystyle{ f:X\to\mathbb{C} }[/math] continuous and [math]\displaystyle{ \mu }[/math] the Lebesgue measure, then [math]\displaystyle{ \operatorname{ess.im}(f)=\overline{\operatorname{im}(f)} }[/math] 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 [math]\displaystyle{ f : X \to Y }[/math], where [math]\displaystyle{ Y }[/math] is a separable metric space. If [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are differentiable manifolds of the same dimension, if [math]\displaystyle{ f\in }[/math] VMO[math]\displaystyle{ (X, Y) }[/math] and if [math]\displaystyle{ \operatorname{ess.im} (f) \ne Y }[/math], then [math]\displaystyle{ \deg f = 0 }[/math].[13]

See also

References

  1. ↑ Zimmer, Robert J. (1990). Essential Results of Functional Analysis. University of Chicago Press. p. 2. ISBN 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. 
  3. ↑ Kon, Mark A. (1985). Probability Distributions in Quantum Statistical Mechanics. Springer. pp. 74, 84. ISBN 3-540-15690-9. 
  4. ↑ Driver, Bruce (May 7, 2012). Analysis Tools with Examples. p. 327. https://mathweb.ucsd.edu/~bdriver/240C-S2018/Lecture_Notes/2012%20Notes/240Lecture_Notes_Ver8.pdf.  Cf. Exercise 30.5.1.
  5. ↑ Segal, Irving E.; Kunze, Ray A. (1978). Integrals and Operators (2nd revised and enlarged ed.). Springer. p. 106. ISBN 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. 
  7. ↑ Weaver, Nik (2013). Measure Theory and Functional Analysis. World Scientific. p. 142. ISBN 978-981-4508-56-8. 
  8. ↑ Bhatia, Rajendra (2009). Notes on Functional Analysis. Hindustan Book Agency. p. 149. ISBN 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. 
  10. ↑ Cf. Tao, Terence (2012). Topics in Random Matrix Theory. American Mathematical Society. p. 29. ISBN 978-0-8218-7430-1. 
  11. ↑ Cf. Freedman, David (1971). Markov Chains. Holden-Day. p. 1. 
  12. ↑ Cf. Chung, Kai Lai (1967). Markov Chains with Stationary Transition Probabilities. Springer. p. 135. 
  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. 



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