Contents

Essential range

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 rangeof${\displaystyle f}$ 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\}.} ^{[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 supportof${\displaystyle f_{*}\mu .}$^[4]

Essential values[edit]

We sometimes use the phrase "essential valueof${\displaystyle f}$" 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[edit]

Y = C[edit]

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[edit]

Say ${\displaystyle (Y,{\cal {T}})}$isdiscrete, i.e., ${\displaystyle {\cal {T}}={\cal {P}}($Y)} is the power setof${\displaystyle Y,}$ i.e., the discrete topologyon${\displaystyle Y.}$ Then the essential range of f is the set of values yinY 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[edit]

• 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[edit]

• 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[edit]

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[edit]

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

References[edit]

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