Essential range

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

Let f be a Borel-measurable, complex-valued function defined on a measure space ${\displaystyle (X,{\mathfrak {A}},\mu )}$. Then the essential range of f is defined to be the set:

${\displaystyle \operatorname {ess.im} (f)=\left\{z\in \mathbb {C} \mid {\text{for all}}\ \varepsilon >0:\mu (\{x:|f(x)-z|<\varepsilon \})>0\right\}}$

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

• 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 ${\displaystyle {\overline {\operatorname {im} (f)}}}$.
• The essential image cannot be used to distinguish functions that are almost everywhere equal: If ${\displaystyle f=g}$ holds ${\displaystyle \mu }$-almost everywhere, then ${\displaystyle \operatorname {ess.im} (f)=\operatorname {ess.im} (g)}$.
• These two facts characterise the essential image: It is the biggest set contained in the closures of ${\displaystyle \operatorname {im} (g)}$ for all g that are a.e. equal to f:

${\displaystyle \operatorname {ess.im} (f)=\bigcap _{f=g\,{\text{a.e.}}}{\overline {\operatorname {im} (g)}}}$.

• The essential range satisfies ${\displaystyle \forall A\subseteq X:f(A)\cap \operatorname {ess.im} (f)=\emptyset \implies \mu (A)=0}$.
• This fact characterises the essential image: It is the smallest closed subset of ${\displaystyle \mathbb {C} }$ 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 ${\displaystyle \sigma (f)}$ where f is considered as an element of the C*-algebra ${\displaystyle L^{\infty }(\mu )}$.

Examples

• If ${\displaystyle \mu }$ 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 ${\displaystyle X\subseteq \mathbb {R} ^{n}}$ is open, ${\displaystyle f:X\to \mathbb {C} }$ continuous and ${\displaystyle \mu }$ the Lebesgue measure, then ${\displaystyle \operatorname {ess.im} (f)={\overline {\operatorname {im} (f)}}}$ holds. This holds more generally for all Borel measures that assign non-zero measure to every non-empty open set.

See also

• Essential supremum and essential infimum
• measure
• L^p space

References

• Walter Rudin (1974). Real and Complex Analysis (2nd ed.). McGraw-Hill. ISBN 978-0-07-054234-1.