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 $(X,,\mu)$ be a measure space, and let $(Y,)$ be a topological space. For any $(,\sigma())$-measurable $f:X\to Y$, we say the essential range of $f$ to mean the set
:$\operatorname(f) = \left\.$
Equivalently, $\operatorname(f)=\operatorname(f_*\mu)$, where $f_*\mu$ is the pushforward measure onto $\sigma()$ of $\mu$ under $f$ and $\operatorname(f_*\mu)$ 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 $(Y,)$ is $\mathbb C$ equipped with its usual topology. Then the essential range of ''f'' is given by
:$\operatorname(f) = \left\.$

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 $(Y,)$ is discrete, i.e., $=(Y)$ 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:
:$\operatorname(f)=\=\.$

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 $\overline$.
* The essential image cannot be used to distinguish functions that are almost everywhere equal: If $f=g$ holds $\mu$-almost everywhere, then $\operatorname(f)=\operatorname(g)$.
* These two facts characterise the essential image: It is the biggest set contained in the closures of $\operatorname(g)$ for all g that are a.e. equal to f:
::$\operatorname(f) = \bigcap_ \overline$.
* The essential range satisfies $\forall A\subseteq X: f(A) \cap \operatorname(f) = \emptyset \implies \mu(A) = 0$.
* This fact characterises the essential image: It is the ''smallest'' closed subset of $\mathbb$ 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 $L^\infty(\mu)$.

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 $X\subseteq\mathbb^n$ is open, $f:X\to\mathbb$ continuous and $\mu$ the Lebesgue measure, then $\operatorname(f)=\overline$ 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 $\operatorname (f) \ne Y$, then $\deg f = 0$.

See also

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

References

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{{DEFAULTSORT:Essential Range

Real analysis
Measure theory