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'.

Let ${\displaystyle (X,{\cal {A}},\mu )}$ be a measure space, and let ${\displaystyle (Y,{\cal {T}})}$ be a topological space. For any ${\displaystyle ({\cal {A}},\sigma ({\cal {T}}))}$-measurable function ${\displaystyle f:X\to Y}$, we say the essential range of ${\displaystyle f}$ to mean the set

Equivalently, ${\displaystyle \operatorname {ess.im} (f)=\operatorname {supp} (f_{*}\mu )}$, where ${\displaystyle f_{*}\mu }$ is the pushforward measure onto ${\displaystyle \sigma ({\cal {T}})}$ of ${\displaystyle \mu }$ under ${\displaystyle f}$ and ${\displaystyle \operatorname {supp} (f_{*}\mu )}$ denotes the support of ${\displaystyle f_{*}\mu .}$

The phrase "essential value of ${\displaystyle f}$" is sometimes used to mean an element of the essential range of ${\displaystyle f.}$

Say ${\displaystyle (Y,{\cal {T}})}$ is ${\displaystyle \mathbb {C} }$ 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.

Say ${\displaystyle (Y,{\cal {T}})}$ is discrete, i.e., ${\displaystyle {\cal {T}}={\cal {P}}(Y)}$ is the power set of ${\displaystyle Y,}$ i.e., the discrete topology on ${\displaystyle Y.}$ Then the essential range of f is the set of values y in Y with strictly positive ${\displaystyle f_{*}\mu }$-measure:

The notion of essential range can be extended to the case of ${\displaystyle f:X\to Y}$, where ${\displaystyle Y}$ is a separable metric space. If ${\displaystyle X}$ and ${\displaystyle Y}$ are differentiable manifolds of the same dimension, if ${\displaystyle f\in }$ VMO${\displaystyle (X,Y)}$ and if ${\displaystyle \operatorname {ess.im} (f)\neq Y}$, then ${\displaystyle \deg f=0}$.