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

${\displaystyle \operatorname {ess.im} (f)=\left\{y\in Y\mid 0<\mu (f^{-1}(U)){\text{ for all }}U\in {\cal {T}}{\text{ with }}y\in U\right\}.}${\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, ${\displaystyle \operatorname {ess.im} (f)=\operatorname {supp} (f_{*}\mu )}${\displaystyle \operatorname {ess.im} (f)=\operatorname {supp} (f_{*}\mu )}, where ${\displaystyle f_{*}\mu }${\displaystyle f_{*}\mu } is the pushforward measure onto ${\displaystyle \sigma ({\cal {T}})}${\displaystyle \sigma ({\cal {T}})} of ${\displaystyle \mu }${\displaystyle \mu } under ${\displaystyle f}${\displaystyle f} and ${\displaystyle \operatorname {supp} (f_{*}\mu )}${\displaystyle \operatorname {supp} (f_{*}\mu )} denotes the support of ${\displaystyle f_{*}\mu .}${\displaystyle f_{*}\mu .}^[4]

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

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

${\displaystyle \operatorname {ess.im} (f)=\left\{z\in \mathbb {C} \mid {\text{for all}}\ \varepsilon \in \mathbb {R} _{>0}:0<\mu \{x\in X:|f(x)-z|<\varepsilon \}\right\}.}${\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 ${\displaystyle (Y,{\cal {T}})}${\displaystyle (Y,{\cal {T}})} is discrete, i.e., ${\displaystyle {\cal {T}}={\cal {P}}(Y)}${\displaystyle {\cal {T}}={\cal {P}}(Y)} is the power set of ${\displaystyle Y,}${\displaystyle Y,} i.e., the discrete topology on ${\displaystyle Y.}${\displaystyle Y.} Then the essential range of f is the set of values y in Y with strictly positive ${\displaystyle f_{*}\mu }${\displaystyle f_{*}\mu }-measure:

${\displaystyle \operatorname {ess.im} (f)=\{y\in Y:0<\mu (f^{\text{pre}}\{y\})\}=\{y\in Y:0<(f_{*}\mu )\{y\}\}.}${\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 ${\displaystyle {\overline {\operatorname {im} (f)}}}${\displaystyle {\overline {\operatorname {im} (f)}}}.
• The essential image cannot be used to distinguish functions that are almost everywhere equal: If ${\displaystyle f=g}${\displaystyle f=g} holds ${\displaystyle \mu }${\displaystyle \mu }-almost everywhere, then ${\displaystyle \operatorname {ess.im} (f)=\operatorname {ess.im} (g)}${\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 ${\displaystyle \operatorname {im} (g)}${\displaystyle \operatorname {im} (g)} for all g that are a.e. equal to f:

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

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

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

• Real analysis
• Measure theory

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