Essential range

In

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 function ${\displaystyle f:X\to Y}$, we say the essential range of ${\displaystyle f}$ 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 support of ${\displaystyle f_{*}\mu .}$^[4]

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

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

${\displaystyle \operatorname {ess.im} (f)=\{y\in Y:0<\mu (f^{\text{pre}}\{y\})\}=\{y\in Y:0<(f_{*}\mu )\{y\}\}.}$^{[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 ${\displaystyle {\overline {\operatorname {im} (f)}}}$.
• 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 ${\displaystyle L^{\infty }(\mu )}$.

• 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.

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 ${\displaystyle \deg f=0}$.^[15]

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