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Wikipedia / 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

be a measure space be a topological space-measurable function, we say the essential range to mean the set

is the pushforward measure denotes the support

Essential values

The phrase "essential value

Special cases of common interest

Y = 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.

(Y,T) is discrete

is discrete is the power set i.e., the discrete topology Then the essential range of f is the set of values y in Y-measure:

Properties

  • The essential range of a measurable function, being the support of a measure, is always closed.

  • .

  • -almost everywhere.

  • for all g that are a.e. equal to f:

    1. .

    .

  • This fact characterises the essential image: It is the smallest 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

  • 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.

  • the Lebesgue measure holds. This holds more generally for all Borel measures that assign non-zero measure to every non-empty open set.

Extension

is a separable metric space are differentiable manifolds VMO.

See also

References