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.
Let f be a Borel-measurable, complex-valued function defined on a measure space ( X , A , μ ) . Then the essential range of f is defined to be the set:
e s s . i m ( f ) = { z ∈ C ∣ for all ε > 0 : μ ( { x : | f ( x ) − z | < ε } ) > 0 } 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.
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 im ( f ) ¯ .The essential image cannot be used to distinguished functions that are almost everywhere equal: If f = g holds μ -almost everywhere, then e s s . i m ( f ) = e s s . i m ( g ) .These two facts characterise the essential image: It is the biggest set contained in the closures of im ( g ) for all g that are a.e. equal to f:The essential range satisfies ∀ A ⊆ X : f ( A ) ∩ e s s . i m ( f ) = ∅ ⟹ μ ( A ) = 0 .This fact characterises the essential image: It is the smallest closed subset of C 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 σ ( f ) where f is considered as an element of the C*-algebra L ∞ ( μ ) .If μ 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 X ⊆ R n is open, f : X → C and μ the Lebesgue measure, then e s s . i m ( f ) = im ( f ) ¯ holds. This holds more generally for all Borel measures that assign non-zero measure to every non-empty open set.