Essential range

Formal definition

Properties

• The essential range of a measurable function is always closed.
• The essential range ess.im 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.