Essential range explained

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,{\calA},\mu)

be a measure space, and let

(Y,{\calT})

be a topological space. For any

({\calA},\sigma({\calT}))

-measurable

f:X\toY

, we say the essential range of

f

to mean the set

\operatorname{ess.im}(f)=\left\{y\inY\mid0<\mu(f-1(U))forallU\in{\calT}withy\inU\right\}.

[1] [2] [3] Equivalently,

\operatorname{ess.im}(f)=\operatorname{supp}(f*\mu)

, where

f*\mu

is the pushforward measure onto

\sigma({\calT})

of

\mu

under

f

and

\operatorname{supp}(f*\mu)

denotes the support of

f*\mu.

[4]

Essential values

We sometimes use the phrase "essential value of

f

" to mean an element of the essential range of

f.

[5] [6]

Special cases of common interest

Y = C

Say

(Y,{\calT})

is

C

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

\operatorname{ess.im}(f)=\left\{z\inC\midforall\varepsilon\inR>0:0<\mu\{x\inX:|f(x)-z|<\varepsilon\}\right\}.

[7] [8] [9]

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,{\calT})

is discrete, i.e.,

{\calT}={\calP}(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\inY:0<\mu(fpre\{y\})\}=\{y\inY:0<(f*\mu)\{y\}\}.

[10] [11] [12]

Properties

\overline{\operatorname{im}(f)}

.

f=g

holds

\mu

-almost everywhere, then

\operatorname{ess.im}(f)=\operatorname{ess.im}(g)

.

\operatorname{im}(g)

for all g that are a.e. equal to f:

\operatorname{ess.im}(f)=capf=ga.e.\overline{\operatorname{im}(g)}

.

\forallA\subseteqX:f(A)\cap\operatorname{ess.im}(f)=\emptyset\implies\mu(A)=0

.

C

with this property.

\sigma(f)

where f is considered as an element of the C*-algebra

Linfty(\mu)

.

Examples

\mu

is the zero measure, then the essential image of all measurable functions is empty.

X\subseteqRn

is open,

f:X\toC

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\toY

, 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)\neY

, then

\degf=0

.[13]

See also

References

Notes and References

  1. Book: Zimmer . Robert J. . Robert Zimmer . Essential Results of Functional Analysis . 1990 . University of Chicago Press . 0-226-98337-4 . 2.
  2. Book: Kuksin . Sergei . Sergei B. Kuksin . Shirikyan . Armen . 2012 . Mathematics of Two-Dimensional Turbulence . Cambridge University Press . 978-1-107-02282-9 . 292.
  3. Book: Kon . Mark A. . Probability Distributions in Quantum Statistical Mechanics . 1985 . Springer . 3-540-15690-9 . 74, 84.
  4. Book: Driver . Bruce . Analysis Tools with Examples . May 7, 2012 . 327 . Cf. Exercise 30.5.1.
  5. Book: Segal . Irving E. . Irving Segal . Kunze . Ray A. . Ray Kunze . Integrals and Operators . 1978 . Springer . 0-387-08323-5 . 106 . 2nd revised and enlarged.
  6. Book: Bogachev . Vladimir I. . Smolyanov . Oleg G. . Real and Functional Analysis . 2020 . Springer . 978-3-030-38219-3 . Moscow Lectures . 2522-0314 . 283.
  7. Book: Weaver . Nik . 2013 . Measure Theory and Functional Analysis . World Scientific . 978-981-4508-56-8 . 142.
  8. Book: Bhatia . Rajendra . Rajendra Bhatia . Notes on Functional Analysis . 2009 . Hindustan Book Agency . 978-81-85931-89-0 . 149.
  9. Book: Folland . Gerald B. . Gerald Folland . Real Analysis: Modern Techniques and Their Applications . 1999 . Wiley . 0-471-31716-0 . 187.
  10. Cf. Book: Tao . Terence . Terence Tao . Topics in Random Matrix Theory . 2012 . American Mathematical Society . 978-0-8218-7430-1 . 29.
  11. Cf. Book: Freedman . David . David A. Freedman . Markov Chains . 1971 . Holden-Day . 1.
  12. Cf. Book: Chung . Kai Lai . Chung Kai-lai . Markov Chains with Stationary Transition Probabilities . 1967 . Springer . 135.
  13. Brezis . Haïm . Nirenberg . Louis . Degree theory and BMO. Part I: Compact manifolds without boundaries . Selecta Mathematica . September 1995 . 1 . 2 . 197–263 . 10.1007/BF01671566.