essential range

Let $(X,\{\backslash cal\; A\},\backslash mu)$ be a measure space, and let $(Y,\{\backslash cal\; T\})$ be a topological space. For any $(\{\backslash cal\; A\},\backslash sigma(\{\backslash cal\; T\}))$-measurable function $f:X\backslash to\; Y$, we say the essential range of $f$ to mean the set

:{{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}}{{rp|at=Example 0.A.5}}{{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}}{{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}}

Equivalently, $\backslash operatorname\{ess.im\}(f)=\backslash operatorname\{supp\}(f\_*\backslash mu)$, where $f\_*\backslash mu$ is the pushforward measure onto $\backslash sigma(\{\backslash cal\; T\})$ of $\backslash mu$ under $f$ and $\backslash operatorname\{supp\}(f\_*\backslash mu)$ denotes the support of $f\_*\backslash mu.${{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.

The phrase "essential value of $f$" is sometimes used to mean an element of the essential range of $f.${{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}}{{rp|at=Exercise 4.1.6}}{{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}}{{rp|at=Example 7.1.11}}

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

:{{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}}{{rp|at=Definition 4.36}}{{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}}{{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}}{{rp|at=cf. Exercise 6.11}}{{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}}{{rp|at=Exercise 3.19}}{{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}}{{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.

Say $(Y,\{\backslash cal\; T\})$ is discrete, i.e., $\{\backslash cal\; T\}=\{\backslash cal\; P\}(Y)$ 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 $f\_*\backslash mu$-measure:

: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}}{{rp|at=Example 1.1.29}}Cf. {{cite book |last1=Freedman |first1=David |author1-link=David A. Freedman |title=Markov Chains |date=1971 |publisher=Holden-Day |page=1}}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}}

• 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 $\backslash overline\{\backslash operatorname\{im\}(f)\}$.
• The essential image cannot be used to distinguish functions that are almost everywhere equal: If $f=g$ holds $\backslash mu$-almost everywhere, then $\backslash operatorname\{ess.im\}(f)=\backslash operatorname\{ess.im\}(g)$.
• These two facts characterise the essential image: It is the biggest set contained in the closures of $\backslash operatorname\{im\}(g)$ for all g that are a.e. equal to f:

::$\backslash operatorname\{ess.im\}(f)\; =\; \backslash bigcap\_\{f=g\backslash ,\backslash text\{a.e.\}\}\; \backslash overline\{\backslash operatorname\{im\}(g)\}$.

• The essential range satisfies $\backslash forall\; A\backslash subseteq\; X:\; f(A)\; \backslash cap\; \backslash operatorname\{ess.im\}(f)\; =\; \backslash emptyset\; \backslash implies\; \backslash mu(A)\; =\; 0$.
• This fact characterises the essential image: It is the smallest closed subset of $\backslash mathbb\{C\}$ 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 $\backslash sigma(f)$ where f is considered as an element of the C*-algebra $L^\backslash infty(\backslash mu)$.

• If $\backslash mu$ 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 $X\backslash subseteq\backslash mathbb\{R\}^n$ is open, $f:X\backslash to\backslash mathbb\{C\}$ continuous and $\backslash mu$ the Lebesgue measure, then $\backslash operatorname\{ess.im\}(f)=\backslash overline\{\backslash operatorname\{im\}(f)\}$ holds. This holds more generally for all Borel measures that assign non-zero measure to every non-empty open set.

The notion of essential range can be extended to the case of $f\; :\; X\; \backslash to\; Y$, where $Y$ is a separable metric space.

If $X$ and $Y$ are differentiable manifolds of the same dimension, if $f\backslash in$ VMO$(X,\; Y)$ and if $\backslash operatorname\{ess.im\}\; (f)\; \backslash ne\; Y$, then $\backslash deg\; f\; =\; 0$.{{cite journal |last1=Brezis |first1=Haïm |last2=Nirenberg |first2=Louis |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}}

• Essential supremum and essential infimum
• measure
• L^p space

{{reflist}}

• {{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

}}

{{Measure theory}}

{{DEFAULTSORT:Essential Range}}

Category:Real analysis

Category:Measure theory