Essential range

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Concept in measure theory

title: "Essential range" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["real-analysis", "measure-theory"] description: "Concept in measure theory" topic_path: "general/real-analysis" source: "https://en.wikipedia.org/wiki/Essential_range" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 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 (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\mid0 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.

Essential values

The phrase "essential value of f" is sometimes used to mean an element of the essential range of f.

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

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

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) \ne Y, then \deg f = 0.

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

References

  1. (1990). "Essential Results of Functional Analysis". University of Chicago Press.
  2. (2012). "Mathematics of Two-Dimensional Turbulence". Cambridge University Press.
  3. (1985). "Probability Distributions in Quantum Statistical Mechanics". Springer.
  4. (May 7, 2012). "Analysis Tools with Examples".
  5. (1978). "Integrals and Operators". Springer.
  6. (2020). "Real and Functional Analysis". Springer.
  7. (2013). "Measure Theory and Functional Analysis". World Scientific.
  8. (2009). "Notes on Functional Analysis". Hindustan Book Agency.
  9. (1999). "Real Analysis: Modern Techniques and Their Applications". Wiley.
  10. (1987). "Real and complex analysis". McGraw-Hill.
  11. (1998). "Banach algebra techniques in operator theory". Springer.
  12. (2012). "Topics in Random Matrix Theory". American Mathematical Society.
  13. (1971). "Markov Chains". Holden-Day.
  14. (1967). "Markov Chains with Stationary Transition Probabilities". Springer.
  15. (September 1995). "Degree theory and BMO. Part I: Compact manifolds without boundaries". Selecta Mathematica.

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