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]