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 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]}

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 }

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.

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]}

- The essential range of a measurable function, being the support of a measure, is always closed.
- The essential range ess.im(f) of a measurable function is always a subset of .
- The essential image cannot be used to distinguish functions that are almost everywhere equal: If holds -almost everywhere, then .
- These two facts characterise the essential image: It is the biggest set contained in the closures of for all g that are a.e. equal to f:

- .

- The essential range satisfies .
- This fact characterises the essential image: It is the
*smallest*closed subset of with this property. - The essential supremum of a real valued function equals the supremum of its essential image and the essential infimum equals the infimum of its essential range. Consequently, a function is essentially bounded if and only if its essential range is bounded.
- The essential range of an essentially bounded function f is equal to the spectrum where f is considered as an element of the C*-algebra .

- If is the zero measure, then the essential image of all measurable functions is empty.
- This also illustrates that even though the essential range of a function is a subset of the closure of the range of that function, equality of the two sets need not hold.
- If is open, continuous and the Lebesgue measure, then holds. This holds more generally for all Borel measures that assign non-zero measure to every non-empty open set.

The notion of essential range can be extended to the case of , where is a separable metric space.
If and are differentiable manifolds of the same dimension, if VMO and if , then .^{[13]}