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Essential range - Wikipedia

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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'.

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 [2][3]}

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]

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}

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 [8][9]: cf. Exercise 6.11 [10]: Exercise 3.19 [11]: 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 , 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\}\}.} ^{[12]: Example 1.1.29 [13][14]}

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 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} .^[15]

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

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. ^ Rudin, Walter (1987). Real and complex analysis (3rd ed.). New York: McGraw-Hill. ISBN 0-07-054234-1.
11. ^ Douglas, Ronald G. (1998). Banach algebra techniques in operator theory (2nd ed.). New York Berlin Heidelberg: Springer. ISBN 0-387-98377-5.
12. ^ Cf. Tao, Terence (2012). Topics in Random Matrix Theory. American Mathematical Society. p. 29. ISBN 978-0-8218-7430-1.
13. ^ Cf. Freedman, David (1971). Markov Chains. Holden-Day. p. 1.
14. ^ Cf. Chung, Kai Lai (1967). Markov Chains with Stationary Transition Probabilities. Springer. p. 135.
15. ^ 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.

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