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 $(X,{\mathfrak {A}},\mu )$. Then the essential range of f is defined to be the set:

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

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

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

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

Examples

• If $\mu$ 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 $X\subseteq \mathbb {R} ^{n}$ is open, $f:X\to \mathbb {C}$ continuous and $\mu$ the Lebesgue measure, then $\operatorname {ess.im} (f)={\overline {\operatorname {im} (f)}}$ 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.