In
mathematics, particularly
measure theory, the essential range, or the set of essential values, of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
is intuitively the 'non-negligible' range of the function: It does not change between two functions that are equal
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
. One way of thinking of the essential range of a function is the
set on which the range of the function is 'concentrated'.
Formal definition
Let
be a measure space, and let
be a topological space. For any
-measurable
, we say the essential range of
to mean the set
:
Equivalently,
, where
is the pushforward measure onto
of
under
and
denotes the
support of
Essential values
We sometimes use the phrase "essential value of
" to mean an element of the essential range of
Special cases of common interest
''Y'' = C
Say
is
equipped with its usual topology. Then the essential range of ''f'' is given by
:
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.
(''Y'',''T'') is discrete
Say
is
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
*Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a g ...
, i.e.,
is the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of
i.e., the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
on
Then the essential range of ''f'' is the set of values ''y'' in ''Y'' with strictly positive
-measure:
:
Properties
* The essential range of a measurable function, being the
support of a measure, 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 distinguish functions that are almost everywhere equal: If
holds
-
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
, 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
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''all' ...
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
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
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.
Extension
The notion of essential range can be extended to the case of
, where
is a separable metric space.
If
and
are differentiable manifolds of the same dimension, if
VMO and if
, then
.
See also
*
Essential supremum and essential infimum
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''all' ...
*
measure
*
Lp space
References
*
{{DEFAULTSORT:Essential Range
Real analysis
Measure theory