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'.

Formal definition

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

$\mathrm{ess.im}(f)=\{y\in Y\mid 0<\mu (f^{-1}(U))\text{ for all }U\in {𝒯}\text{ with }y\in U\}.$

Equivalently, $\mathrm{ess.im}(f)=\mathrm{supp}(f_{*}\mu )$, where $f_{*}\mu$ is the pushforward measure onto $\sigma ({𝒯})$ of $\mu$ under $f$ and $\mathrm{supp}(f_{*}\mu )$ denotes the support of $f_{*}\mu .$

Essential values

We sometimes use the phrase "essential value of $f$" to mean an element of the essential range of $f.$

Special cases of common interest

Y = C

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

$\mathrm{ess.im}(f)=\{z\in \mathbb{ℂ}\mid \text{for all}\epsilon \in {\mathbb{ℝ}}_{>0}:0<\mu \{x\in X:|f(x)-z|<\epsilon \}\}.$

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

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

$\mathrm{ess.im}(f)=\{y\in Y:0<\mu (f^{\text{pre}}\{y\})\}=\{y\in Y:0<(f_{*}\mu )\{y\}\}.$

Properties

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

$\mathrm{ess.im}(f)=\bigcap _{f=g\text{a.e.}}\overline{\mathrm{im}(g)}$.

• The essential range satisfies $\mathrm{\forall }A\subseteq X:f(A)\cap \mathrm{ess.im}(f)=\mathrm{\varnothing }⟹\mu (A)=0$.
• This fact characterises the essential image: It is the smallest closed subset of $\mathbb{ℂ}$ 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 $\sigma (f)$ where f is considered as an element of the C*-algebra $L^{\mathrm{\infty }}(\mu )$.

Examples

• If $\mu$ 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 $X\subseteq {\mathbb{ℝ}}^n$ is open, $f:X\to \mathbb{ℂ}$ continuous and $\mu$ the Lebesgue measure, then $\mathrm{ess.im}(f)=\overline{\mathrm{im}(f)}$ holds. This holds more generally for all Borel measures that assign non-zero measure to every non-empty open set.

Extension

The notion of essential range can be extended to the case of $f:X\to Y$, where $Y$ is a separable metric space. If $X$ and $Y$ are differentiable manifolds of the same dimension, if $f\in$ VMO$(X,Y)$ and if $\mathrm{ess.im}(f)\ne Y$, then $\mathrm{deg}f=0$.

See also

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

References

• Walter Rudin (1974). Real and Complex Analysis (2nd ed.). McGraw-Hill. ISBN 978-0-07-054234-1. https://archive.org/details/realcomplexanaly00rudi_0.