Английская Википедия:Essential range

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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'.

Formal definition

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

<math>\operatorname{ess.im}(f) = \left\{y\in Y\mid0<\mu(f^{-1}(U))\text{ for all }U\in{\cal T} \text{ with } y \in U\right\}.</math>Шаблон:Rp

Equivalently, <math>\operatorname{ess.im}(f)=\operatorname{supp}(f_*\mu)</math>, where <math>f_*\mu</math> is the pushforward measure onto <math>\sigma({\cal T})</math> of <math>\mu</math> under <math>f</math> and <math>\operatorname{supp}(f_*\mu)</math> denotes the support of <math>f_*\mu.</math>

Essential values

We sometimes use the phrase "essential value of <math>f</math>" to mean an element of the essential range of <math>f.</math>Шаблон:RpШаблон:Rp

Special cases of common interest

Y = C

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

<math>\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\}.</math>Шаблон:RpШаблон:Rp

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

<math>\operatorname{ess.im}(f)=\{y\in Y:0<\mu(f^\text{pre}\{y\})\}=\{y\in Y:0<(f_*\mu)\{y\}\}.</math>Шаблон:Rp

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 <math>\overline{\operatorname{im}(f)}</math>.
• The essential image cannot be used to distinguish functions that are almost everywhere equal: If <math>f=g</math> holds <math>\mu</math>-almost everywhere, then <math>\operatorname{ess.im}(f)=\operatorname{ess.im}(g)</math>.
• These two facts characterise the essential image: It is the biggest set contained in the closures of <math>\operatorname{im}(g)</math> for all g that are a.e. equal to f:

<math>\operatorname{ess.im}(f) = \bigcap_{f=g\,\text{a.e.}} \overline{\operatorname{im}(g)}</math>.

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

Examples

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

See also

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

References

• Шаблон:Cite book