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'.
, μ )
be a measure space, and let
be a topological space.
, we say the essential range of
to mean the set Equivalently,
μ
is the pushforward measure onto
μ
μ )
denotes the support of
μ .
[4] The phrase "essential value of
" is sometimes used to mean an element of the essential range of
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.
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
μ
-measure: The notion of essential range can be extended to the case of
is a separable metric space.
are differentiable manifolds of the same dimension, if