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

${\displaystyle \operatorname {ess.im} (f)=\left\{y\in Y\mid 0<\mu (f^{\operatorname {pre} }U){\text{ for all }}U\in {\cal {T}}\right\}.}$ ^{[1] : Example 0.A.5  [2] [3]}

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

Essential values

We sometimes use the phrase "essential value of ${\displaystyle f}$" to mean an element of the essential range of ${\displaystyle f.}$ ^{[5] : Exercise 4.1.6  [6] : Example 7.1.11}

Special cases of common interest

Y = C

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

${\displaystyle \operatorname {ess.im} (f)=\left\{z\in \mathbb {C} \mid {\text{for all}}\ \varepsilon \in \mathbb {R} _{>0}:0<\mu \{x\in X:|f(x)-z|<\varepsilon \}\right\}.}$ ^{[7] : Definition 4.36  [8] [9] : cf. Exercise 6.11}

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 ${\displaystyle (Y,{\cal {T}})}$ is discrete, i.e., ${\displaystyle {\cal {T}}={\cal {P}}(Y)}$ is the power set of ${\displaystyle Y,}$ i.e., the discrete topology on ${\displaystyle Y.}$ Then the essential range of f is the set of values y in Y with strictly positive ${\displaystyle f_{*}\mu }$-measure:

${\displaystyle \operatorname {ess.im} (f)=\{y\in Y:0<\mu (f^{\text{pre}}\{y\})\}=\{y\in Y:0<(f_{*}\mu )\{y\}\}.}$ ^{[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 ${\displaystyle {\overline {\operatorname {im} (f)}}}$.
• The essential image cannot be used to distinguish functions that are almost everywhere equal: If ${\displaystyle f=g}$ holds ${\displaystyle \mu }$-almost everywhere, then ${\displaystyle \operatorname {ess.im} (f)=\operatorname {ess.im} (g)}$.
• These two facts characterise the essential image: It is the biggest set contained in the closures of ${\displaystyle \operatorname {im} (g)}$ for all g that are a.e. equal to f:

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

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

Examples

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

See also

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

References

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.

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