Essential range

{{Short description|Concept in measure theory}} 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>(X,{\cal A},\mu)</math> be a measure space, and let <math>(Y,{\cal T})</math> be a topological space. For any <math>({\cal A},\sigma({\cal T}))</math>-measurable function <math>f:X\to Y</math>, we say the '''essential range''' of <math>f</math> to mean the set :<math>\operatorname{ess.im}(f) = \left\{y\in Y\mid0<\mu(f^{-1}(U))\text{ for all }U\in{\cal T} \text{ with } y \in U\right\}.</math><ref>{{cite book |last1=Zimmer |first1=Robert J. |author1-link=Robert Zimmer |title=Essential Results of Functional Analysis |date=1990 |publisher=University of Chicago Press |isbn=0-226-98337-4 |page=2}}</ref>{{rp|at=Example 0.A.5}}<ref>{{cite book |last1=Kuksin |first1=Sergei |author1-link=Sergei B. Kuksin |last2=Shirikyan |first2=Armen |date=2012 |title=Mathematics of Two-Dimensional Turbulence |publisher=Cambridge University Press |isbn=978-1-107-02282-9 |page=292}}</ref><ref>{{cite book |last1=Kon |first1=Mark A. |title=Probability Distributions in Quantum Statistical Mechanics |date=1985 |publisher=Springer |isbn=3-540-15690-9 |pages=74, 84}}</ref> Equivalently, <math>\operatorname{ess.im}(f)=\operatorname{supp}(f_*\mu)</math>, where <math>f_*\mu</math> is the pushforward measure onto <math>\sigma({\cal T})</math> of <math>\mu</math> under <math>f</math> and <math>\operatorname{supp}(f_*\mu)</math> denotes the support of <math>f_*\mu.</math><ref>{{cite book |last1=Driver |first1=Bruce |title=Analysis Tools with Examples |date=May 7, 2012 |page=327 |url=https://mathweb.ucsd.edu/~bdriver/240C-S2018/Lecture_Notes/2012%20Notes/240Lecture_Notes_Ver8.pdf}} Cf. Exercise 30.5.1.</ref>

===Essential values=== The phrase "'''essential value''' of <math>f</math>" is sometimes used to mean an element of the essential range of <math>f.</math><ref>{{cite book |last1=Segal |first1=Irving E. |author1-link=Irving Segal |last2=Kunze |first2=Ray A. |author2-link=Ray Kunze |title=Integrals and Operators |date=1978 |publisher=Springer |isbn=0-387-08323-5 |page=106 |edition=2nd revised and enlarged}}</ref>{{rp|at=Exercise 4.1.6}}<ref>{{cite book |last1=Bogachev |first1=Vladimir I. |last2=Smolyanov |first2=Oleg G. |title=Real and Functional Analysis |date=2020 |publisher=Springer |isbn=978-3-030-38219-3 |series=Moscow Lectures |issn=2522-0314 |page=283}}</ref>{{rp|at=Example 7.1.11}}

==Special cases of common interest== ===''Y'' = '''C'''=== Say <math>(Y,{\cal T})</math> is <math>\mathbb C</math> equipped with its usual topology. Then the essential range of ''f'' is given by :<math>\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\}.</math><ref>{{cite book |last1=Weaver |first1=Nik |date=2013 |title=Measure Theory and Functional Analysis |publisher=World Scientific |isbn=978-981-4508-56-8 |page=142}}</ref>{{rp|at=Definition 4.36}}<ref>{{cite book |last1=Bhatia |first1=Rajendra |author1-link=Rajendra Bhatia |title=Notes on Functional Analysis |date=2009 |publisher=Hindustan Book Agency |isbn=978-81-85931-89-0 |page=149}}</ref><ref>{{cite book |last1=Folland |first1=Gerald B. |author1-link=Gerald Folland |title=Real Analysis: Modern Techniques and Their Applications |date=1999 |publisher=Wiley |isbn=0-471-31716-0 |page=187}}</ref>{{rp|at=cf. Exercise 6.11}}<ref>{{cite book |last1=Rudin |first1=Walter |title=Real and complex analysis |date=1987 |publisher=McGraw-Hill |location=New York |isbn=0-07-054234-1 |edition=3rd}}</ref>{{rp|at=Exercise 3.19}}<ref>{{cite book |last1=Douglas |first1=Ronald G. |title=Banach algebra techniques in operator theory |date=1998 |publisher=Springer |location=New York Berlin Heidelberg |isbn=0-387-98377-5 |edition=2nd}}</ref>{{rp|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 <math>(Y,{\cal T})</math> is discrete, i.e., <math>{\cal T}={\cal P}(Y)</math> is the power set of <math>Y,</math> i.e., the discrete topology on <math>Y.</math> Then the essential range of ''f'' is the set of values ''y'' in ''Y'' with strictly positive <math>f_*\mu</math>-measure: :<math>\operatorname{ess.im}(f)=\{y\in Y:0<\mu(f^\text{pre}\{y\})\}=\{y\in Y:0<(f_*\mu)\{y\}\}.</math><ref>Cf. {{cite book |last1=Tao |first1=Terence |author1-link=Terence Tao |title=Topics in Random Matrix Theory |date=2012 |publisher=American Mathematical Society |isbn=978-0-8218-7430-1 |page=29}}</ref>{{rp|at=Example 1.1.29}}<ref>Cf. {{cite book |last1=Freedman |first1=David |author1-link=David A. Freedman |title=Markov Chains |date=1971 |publisher=Holden-Day |page=1}}</ref><ref>Cf. {{cite book |last1=Chung |first1=Kai Lai |author1-link=Chung Kai-lai |title=Markov Chains with Stationary Transition Probabilities |date=1967 |publisher=Springer |page=135}}</ref>

==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>\overline{\operatorname{im}(f)}</math>. * The essential image cannot be used to distinguish functions that are almost everywhere equal: If <math>f=g</math> holds <math>\mu</math>-almost everywhere, then <math>\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>\operatorname{im}(g)</math> for all g that are a.e. equal to f: ::<math>\operatorname{ess.im}(f) = \bigcap_{f=g\,\text{a.e.}} \overline{\operatorname{im}(g)}</math>. * The essential range satisfies <math>\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>\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>\sigma(f)</math> where f is considered as an element of the C*-algebra <math>L^\infty(\mu)</math>.

== Examples ==

* If <math>\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>X\subseteq\mathbb{R}^n</math> is open, <math>f:X\to\mathbb{C}</math> continuous and <math>\mu</math> the Lebesgue measure, then <math>\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>f : X \to Y</math>, where <math>Y</math> is a separable metric space. If <math>X</math> and <math>Y</math> are differentiable manifolds of the same dimension, if <math>f\in</math> VMO<math>(X, Y)</math> and if <math>\operatorname{ess.im} (f) \ne Y</math>, then <math>\deg f = 0</math>.<ref>{{cite journal |last1=Brezis |first1=Haïm |author-link=Haïm Brezis |last2=Nirenberg |first2=Louis |author-link2=Louis Nirenberg |title=Degree theory and BMO. Part I: Compact manifolds without boundaries |journal=Selecta Mathematica |date=September 1995 |volume=1 |issue=2 |pages=197–263 |doi=10.1007/BF01671566}}</ref>

==References==

* {{cite book | author = Walter Rudin | author-link = Walter Rudin | year = 1974 | title = Real and Complex Analysis | url = https://archive.org/details/realcomplexanaly00rudi_0 | url-access = registration | edition = 2nd | publisher = McGraw-Hill | isbn = 978-0-07-054234-1 }}

Category:Real analysis Category:Measure theory