Essential range

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In mathematics, particularly measure theory, the essential range 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 most 'concentrated'. The essential range can be defined for measurable real or complex-valued functions on a measure space.

Formal definition[edit]

Let f be a Borel-measurable, complex-valued function defined on a measure space . Then the essential range of f is defined to be the set:

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.

Properties[edit]

  • The essential range of a measurable function 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 distinguished 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[edit]

  • 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, 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.

See also[edit]

References[edit]