In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a vector measure is 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-orie ...
defined on a
family of sets
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F of subsets of a given set S is called a family of su ...
and taking
vector
Vector most often refers to:
* Euclidean vector, a quantity with a magnitude and a direction
* Disease vector, an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematics a ...
values satisfying certain properties. It is a generalization of the concept of finite
measure, which takes
nonnegative
In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. ...
real values only.
Definitions and first consequences
Given a
field of sets
In mathematics, a field of sets is a mathematical structure consisting of a pair ( X, \mathcal ) consisting of a set X and a family \mathcal of subsets of X called an algebra over X that contains the empty set as an element, and is closed under t ...
and a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
a finitely additive vector measure (or measure, for short) is a function
such that for any two
disjoint set
In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' wh ...
s
and
in
one has
A vector measure
is called countably additive if for any
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of disjoint sets in
such that their union is in
it holds that
with the
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used i ...
on the right-hand side convergent in the
norm
Norm, the Norm or NORM may refer to:
In academic disciplines
* Normativity, phenomenon of designating things as good or bad
* Norm (geology), an estimate of the idealised mineral content of a rock
* Norm (philosophy), a standard in normative e ...
of the Banach space
It can be proved that an additive vector measure
is countably additive if and only if for any sequence
as above one has
where
is the norm on
Countably additive vector measures defined on
sigma-algebras are more general than finite
measures, finite
signed measure
In mathematics, a signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to acquire sign.
Definition
There are two slightly different concepts of a signed measure, de ...
s, and
complex measure
In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number.
Definition
Formal ...
s, which are
countably additive functions taking values respectively on the real interval
the set of real numbers, and the set of complex number">real_number.html" ;"title=", \infty), the set of real number">, \infty), the set of real numbers, and the set of complex numbers.
Examples
Consider the field of sets made up of the interval
together with the family
of all Lebesgue measurable sets contained in this interval. For any such set
define
where
is the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
of
Depending on where
is declared to take values, two different outcomes are observed.
*
viewed as a function from
to the
-space is a vector measure which is not countably-additive.
*
viewed as a function from
to the
-space
is a countably-additive vector measure.
Both of these statements follow quite easily from the criterion () stated above.
The variation of a vector measure
Given a vector measure
the variation
of
is defined as
where the
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
is taken over all the
partitions
of
into a finite number of disjoint sets, for all
in
Here,
is the norm on
The variation of
is a finitely additive function taking values in
It holds that
for any
in
If
is finite, the measure
is said to be of bounded variation. One can prove that if
is a vector measure of bounded variation, then
is countably additive if and only if
is countably additive.
Lyapunov's theorem
In the theory of vector measures, ''
Lyapunov's theorem'' states that the range of a (
non-atomic) finite-dimensional vector measure is
closed and
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
.
[ Kluvánek, I., Knowles, G., ''Vector Measures and Control Systems'', North-Holland Mathematics Studies 20, Amsterdam, 1976.] In fact, the range of a non-atomic vector measure is a
zonoid
In convex geometry, a zonoid is a type of centrally symmetric convex body.
Definitions
The zonoids have several definitions, equivalent up to translations of the resulting shapes:
* A zonoid is a shape that can be approximated arbitrarily clos ...
(the closed and convex set that is the limit of a convergent sequence of
zonotope
In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...
s).
It is used in
economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interac ...
,
[ This paper builds on two papers by Aumann: ]
in (
"bang–bang")
control theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...
,
and in
statistical theory
The theory of statistics provides a basis for the whole range of techniques, in both study design and data analysis, that are used within applications of statistics.
The theory covers approaches to statistical-decision problems and to statistica ...
.
Lyapunov's theorem has been proved by using the
Shapley–Folkman lemma
The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of set (mathematics), sets in a vector space. The lemma may be intuitively understood as saying that, if the number of summed sets exceeds the dime ...
, which has been viewed as a
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, ...
analogue of Lyapunov's theorem.
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See also
*
*
*
*
*
*
*
References
Bibliography
*
*
*
Kluvánek, I., Knowles, G, ''Vector Measures and Control Systems'', North-Holland Mathematics Studies 20, Amsterdam, 1976.
*
*
{{Measure theory
Control theory
Functional analysis
Measures (measure theory)