Essential range
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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 function , we say the essential range of to mean the set
Equivalently, , where is the pushforward measure onto of under and denotes the support of [4]
Essential values
The phrase "essential value of " is sometimes used to mean an element of the essential range of [5]: Exercise 4.1.6 [6]: Example 7.1.11
Special cases of common interest
Summarize
Perspective
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 .[15]
See also
References
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