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 , we say the essential range of to mean the set
- [1]: Example 0.A.5 [2][3]
Equivalently, , where is the pushforward measure onto of under and denotes the support of [4]
Essential values
We sometimes use the phrase "essential value of " to mean an element of the essential range of [5]: Exercise 4.1.6 [6]: Example 7.1.11
Special cases of common interest
Y = C
Say is equipped with its usual topology. Then the essential range of f is given by
- [7]: Definition 4.36 [8][9]: cf. Exercise 6.11
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 i.e., the discrete topology on Then the essential range of f is the set of values y in Y with strictly positive -measure:
- [10]: Example 1.1.29 [11][12]