Essential range

Formal definition

Let (X,{\cal A},\mu) be a measure space, and let (Y,{\cal T}) be a topological space. For any ({\cal A},\sigma({\cal T}))-measurable function f:X\to Y, we say the essential range of f to mean the set :\operatorname{ess.im}(f) = \left{y\in Y\mid0 Equivalently, \operatorname{ess.im}(f)=\operatorname{supp}(f_\mu), where f_\mu is the pushforward measure onto \sigma({\cal T}) of \mu under f and \operatorname{supp}(f_\mu) denotes the support of f_\mu.

Essential values

The phrase "essential value of f" is sometimes used to mean an element of the essential range of f.

Special cases of common interest

''Y'' = '''C'''

Say (Y,{\cal T}) is \mathbb C equipped with its usual topology. Then the essential range of f is given by :\operatorname{ess.im}(f) = \left{z \in \mathbb{C} \mid \text{for all}\ \varepsilon\in\mathbb R_{0}: 0

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,{\cal T}) is discrete, i.e., {\cal T}={\cal 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: :\operatorname{ess.im}(f)={y\in Y:0

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{\operatorname{im}(f)}.
• The essential image cannot be used to distinguish functions that are almost everywhere equal: If f=g holds \mu-almost everywhere, then \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 \operatorname{im}(g) for all g that are a.e. equal to f: ::\operatorname{ess.im}(f) = \bigcap_{f=g,\text{a.e.}} \overline{\operatorname{im}(g)}.
• The essential range satisfies \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 \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^\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 \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

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{ess.im} (f) \ne Y, then \deg f = 0.

References

• {{cite book | author-link = Walter Rudin | url-access = registration

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