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 be a measure space, and let be a topological space. For any -measurable , we say the essential range of to mean the set

[1]: Example 0.A.5 [2][3]

Equivalently, , where is the pushforward measure onto of under and denotes the support of [4]

Essential values

We sometimes use the phrase "essential value of " to mean an element of the essential range of [5]: Exercise 4.1.6 [6]: Example 7.1.11 

Special cases of common interest

Y = C

Say is equipped with its usual topology. Then the essential range of f is given by

[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 is discrete, i.e., is the power set of i.e., the discrete topology on Then the essential range of f is the set of values y in Y with strictly positive -measure:

[10]: Example 1.1.29 [11][12]





The notion of essential range can be extended to the case of , where is a separable metric space. If and are differentiable manifolds of the same dimension, if VMO and if , then .[13]

See also


  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.