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'.

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

e s s . i m ⁡ ( f ) = { y ∈ Y ∣ 0 < μ ( f − 1 ( U ) )  for all  U ∈ T  with  y ∈ U } . {\displaystyle \operatorname {ess.im} (f)=\left\{y\in Y\mid 0<\mu (f^{-1}(U)){\text{ for all }}U\in {\cal {T}}{\text{ with }}y\in U\right\}.} ^{[1]: Example 0.A.5 [2][3]}

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

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

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

e s s . i m ⁡ ( f ) = { z ∈ C ∣ for all   ε ∈ R > 0 : 0 < μ { x ∈ X : | f ( x ) − z | < ε } } . {\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 [10]: Exercise 3.19 [11]: Definition 2.61}

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.

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

e s s . i m ⁡ ( f ) = { y ∈ Y : 0 < μ ( f pre { y } ) } = { y ∈ Y : 0 < ( f ∗ μ ) { y } } . {\displaystyle \operatorname {ess.im} (f)=\{y\in Y:0<\mu (f^{\text{pre}}\{y\})\}=\{y\in Y:0<(f_{*}\mu )\{y\}\}.} ^{[12]: Example 1.1.29 [13][14]}

• 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 im ⁡ ( f ) ¯ {\displaystyle {\overline {\operatorname {im} (f)}}} .
• The essential image cannot be used to distinguish functions that are almost everywhere equal: If f = g {\displaystyle f=g} holds μ {\displaystyle \mu } -almost everywhere, then e s s . i m ⁡ ( f ) = e s s . i m ⁡ ( g ) {\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 im ⁡ ( g ) {\displaystyle \operatorname {im} (g)} for all g that are a.e. equal to f:

e s s . i m ⁡ ( f ) = ⋂ f = g a.e. im ⁡ ( g ) ¯ {\displaystyle \operatorname {ess.im} (f)=\bigcap _{f=g\,{\text{a.e.}}}{\overline {\operatorname {im} (g)}}} .

• The essential range satisfies ∀ A ⊆ X : f ( A ) ∩ e s s . i m ⁡ ( f ) = ∅ ⟹ μ ( A ) = 0 {\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 C {\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 σ ( f ) {\displaystyle \sigma (f)} where f is considered as an element of the C*-algebra L ∞ ( μ ) {\displaystyle L^{\infty }(\mu )} .

• If μ {\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 X ⊆ R n {\displaystyle X\subseteq \mathbb {R} ^{n}} is open, f : X → C {\displaystyle f:X\to \mathbb {C} } continuous and μ {\displaystyle \mu } the Lebesgue measure, then e s s . i m ⁡ ( f ) = im ⁡ ( f ) ¯ {\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.

The notion of essential range can be extended to the case of f : X → Y {\displaystyle f:X\to Y} , where Y {\displaystyle Y} is a separable metric space. If X {\displaystyle X} and Y {\displaystyle Y} are differentiable manifolds of the same dimension, if f ∈ {\displaystyle f\in } VMO ( X , Y ) {\displaystyle (X,Y)} and if e s s . i m ⁡ ( f ) ≠ Y {\displaystyle \operatorname {ess.im} (f)\neq Y} , then deg ⁡ f = 0 {\displaystyle \deg f=0} .^[15]

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

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