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'.

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,{\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) . 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. 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 . ==See also== Essential supremum and essential infimum measure L p space ==References==

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