Inmathematics, particularlymeasure theory, theessential range, or the set ofessential values, of afunction is intuitively the 'non-negligible' range of the function: It does not change between two functions that are equalalmost everywhere. One way of thinking of the essential range of a function is theset on which the range of the function is 'concentrated'.

Let${\displaystyle (X,\mathcal{A},\mu )}$ be ameasure space, and let${\displaystyle (Y,\mathcal{T})}$ be atopological space. For any${\displaystyle (\mathcal{A},\sigma (\mathcal{T}))}$-measurable function${\displaystyle f:X\to Y}$, we say theessential range of${\displaystyle f}$ to mean the set

^{[1]: Example 0.A.5 [2][3]}

Equivalently,${\displaystyle \mathrm{e}\mathrm{s}\mathrm{s}.\mathrm{i}\mathrm{m}(f)=\mathrm{supp}(f_{\ast }\mu )}$, where${\displaystyle f_{\ast }\mu }$ is thepushforward measure onto${\displaystyle \sigma (\mathcal{T})}$ of${\displaystyle \mu }$ under${\displaystyle f}$ and${\displaystyle \mathrm{supp}(f_{\ast }\mu )}$ denotes thesupport of${\displaystyle f_{\ast }\mu .}$^[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${\displaystyle (Y,\mathcal{T})}$ is${\displaystyle \mathbb{C}}$ equipped with its usual topology. Then the essential range off is given by

^{[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 numbersz such that the inverse image of each ε-neighbourhood ofz underf has positive measure.

Say${\displaystyle (Y,\mathcal{T})}$ isdiscrete, i.e.,${\displaystyle \mathcal{T}=\mathcal{P}(Y)}$ is thepower set of${\displaystyle Y,}$ i.e., thediscrete topology on${\displaystyle Y.}$ Then the essential range off is the set of valuesy inY with strictly positive${\displaystyle f_{\ast }\mu }$-measure:

^{[12]: Example 1.1.29 [13][14]}

• The essential range of a measurable function, being thesupport of a measure, is always closed.
• The essential range ess.im(f) of a measurable function is always a subset of${\displaystyle \overline{\mathrm{im}(f)}}$.
• The essential image cannot be used to distinguish functions that are almost everywhere equal: If${\displaystyle f=g}$ holds${\displaystyle \mu }$-almost everywhere, then${\displaystyle \mathrm{e}\mathrm{s}\mathrm{s}.\mathrm{i}\mathrm{m}(f)=\mathrm{e}\mathrm{s}\mathrm{s}.\mathrm{i}\mathrm{m}(g)}$.
• These two facts characterise the essential image: It is the biggest set contained in the closures of${\displaystyle \mathrm{im}(g)}$ for all g that are a.e. equal to f:

${\displaystyle \mathrm{e}\mathrm{s}\mathrm{s}.\mathrm{i}\mathrm{m}(f)=\bigcap _{f=g\text{a.e.}}\overline{\mathrm{im}(g)}}$.

• The essential range satisfies${\displaystyle \mathrm{\forall }A\subseteq X:f(A)\cap \mathrm{e}\mathrm{s}\mathrm{s}.\mathrm{i}\mathrm{m}(f)=\mathrm{\varnothing }⟹\mu (A)=0}$.
• This fact characterises the essential image: It is thesmallest closed subset of${\displaystyle \mathbb{C}}$ with this property.
• Theessential 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 thespectrum${\displaystyle \sigma (f)}$ where f is considered as an element of theC*-algebra${\displaystyle L^{\mathrm{\infty }}(\mu )}$.

• 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${\displaystyle \mu }$ theLebesgue measure, then${\displaystyle \mathrm{e}\mathrm{s}\mathrm{s}.\mathrm{i}\mathrm{m}(f)=\overline{\mathrm{im}(f)}}$ holds. This holds more generally for allBorel 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 aseparablemetric space.If${\displaystyle X}$ and${\displaystyle Y}$ aredifferentiable manifolds of the same dimension, if${\displaystyle f\in }$VMO${\displaystyle (X,Y)}$ and if${\displaystyle \mathrm{e}\mathrm{s}\mathrm{s}.\mathrm{i}\mathrm{m}(f)\ne Y}$, then${\displaystyle \mathrm{deg}f=0}$.^[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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