Essential range

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

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

Essential values

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

Special cases of common interest

Y = C

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

\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}

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 $(Y,{\cal {T}})$ is discrete, i.e., ${\cal {T}}={\cal {P}}(Y)$ 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 $f_{*}\mu$ -measure:

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

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

.

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

Examples

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

References

^ Zimmer, Robert J. (1990). Essential Results of Functional Analysis. University of Chicago Press. p. 2. ISBN 0-226-98337-4.
^ Kuksin, Sergei; Shirikyan, Armen (2012). Mathematics of Two-Dimensional Turbulence. Cambridge University Press. p. 292. ISBN 978-1-107-02282-9.
^ Kon, Mark A. (1985). Probability Distributions in Quantum Statistical Mechanics. Springer. pp. 74, 84. ISBN 3-540-15690-9.
^ Driver, Bruce (May 7, 2012). Analysis Tools with Examples (PDF). p. 327. Cf. Exercise 30.5.1.
^ Segal, Irving E.; Kunze, Ray A. (1978). Integrals and Operators (2nd revised and enlarged ed.). Springer. p. 106. ISBN 0-387-08323-5.
^ 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.
^ Weaver, Nik (2013). Measure Theory and Functional Analysis. World Scientific. p. 142. ISBN 978-981-4508-56-8.
^ Bhatia, Rajendra (2009). Notes on Functional Analysis. Hindustan Book Agency. p. 149. ISBN 978-81-85931-89-0.
^ Folland, Gerald B. (1999). Real Analysis: Modern Techniques and Their Applications. Wiley. p. 187. ISBN 0-471-31716-0.
^ Cf. Tao, Terence (2012). Topics in Random Matrix Theory. American Mathematical Society. p. 29. ISBN 978-0-8218-7430-1.
^ Cf. Freedman, David (1971). Markov Chains. Holden-Day. p. 1.
^ Cf. Chung, Kai Lai (1967). Markov Chains with Stationary Transition Probabilities. Springer. p. 135.
^ 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.

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

[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 (PDF). p. 327. 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. ISSN 2522-0314.

[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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