Essential range

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

Equivalently, , where is the pushforward measure onto of under and denotes the support of

Essential values

We sometimes use the phrase "essential value of " to mean an element of the essential range of

Special cases of common interest

Y = C

Say is 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.

(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:

Properties

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 .

Examples

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.

Extension

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 .