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 be a measure space, and let be a topological space. For any -measurable function f:X\to Y, we say the essential range of f to mean the set Equivalently,, where f_\mu is the pushforward measure onto of \mu under f and 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 is \mathbb C equipped with its usual topology. Then the essential range of f is given by 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 is discrete, i.e., 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:

Properties

Examples

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, then \deg f = 0.