HOME

TheInfoList



OR:

In mathematics, especially
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, a set function is a function whose domain is a
family Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...
of
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s \R and \pm \infty. A set function generally aims to subsets in some way. Measures are typical examples of "measuring" set functions. Therefore, the term "set function" is often used for avoiding confusion between the mathematical meaning of "measure" and its common language meaning.


Definitions

If \mathcal is 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 ...
over \Omega (meaning that \mathcal \subseteq \wp(\Omega) where \wp(\Omega) denotes the powerset) then a is a function \mu with domain \mathcal and codomain \infty, \infty/math> or, sometimes, the codomain is instead some
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
, as with vector measures, complex measures, and projection-valued measures. The domain of a set function may have any number properties; the commonly encountered properties and categories of families are listed in the table below. In general, it is typically assumed that \mu(E) + \mu(F) is always well-defined for all E, F \in \mathcal, or equivalently, that \mu does not take on both - \infty and + \infty as values. This article will henceforth assume this; although alternatively, all definitions below could instead be qualified by statements such as "whenever the sum/series is defined". This is sometimes done with subtraction, such as with the following result, which holds whenever \mu is finitely additive: :: \mu(F) - \mu(E) = \mu(F \setminus E) \text \mu(F) - \mu(E) is defined with E, F \in \mathcal satisfying E \subseteq F and F \setminus E \in \mathcal. Null sets A set F \in \mathcal is called a (with respect to \mu) or simply if \mu(F) = 0. Whenever \mu is not identically equal to either -\infty or +\infty then it is typically also assumed that: Variation and mass The S is , \mu, (S) ~\stackrel~ \sup \ where , \,\cdot\,, denotes the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
(or more generally, it denotes the norm or
seminorm In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
if \mu is vector-valued in a ( semi) normed space). Assuming that \cup \mathcal ~\stackrel~ \textstyle\bigcup\limits_ F \in \mathcal, then , \mu, \left(\cup \mathcal\right) is called the of \mu and \mu\left(\cup \mathcal\right) is called the of \mu. A set function is called if for every F \in \mathcal, the value \mu(F) is (which by definition means that \mu(F) \neq \infty and \mu(F) \neq -\infty; an is one that is equal to \infty or - \infty). Every finite set function must have a finite
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
.


Common properties of set functions

A set function \mu on \mathcal is said to be Arbitrary sums As described in this article's section on generalized series, for any family \left(r_i\right)_ of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s indexed by an arbitrary indexing set I, it is possible to define their sum \textstyle\sum\limits_ r_i as the limit of the net of finite partial sums F \in \operatorname(I) \mapsto \textstyle\sum\limits_ r_i where the domain \operatorname(I) is
directed Direct may refer to: Mathematics * Directed set, in order theory * Direct limit of (pre), sheaves * Direct sum of modules, a construction in abstract algebra which combines several vector spaces Computing * Direct access (disambiguation), a ...
by \,\subseteq.\, Whenever this net converges then its limit is denoted by the symbols \textstyle\sum\limits_ r_i while if this net instead diverges to \pm \infty then this may be indicated by writing \textstyle\sum\limits_ r_i = \pm \infty. Any sum over the empty set is defined to be zero; that is, if I = \varnothing then \textstyle\sum\limits_ r_i = 0 by definition. For example, if z_i = 0 for every i \in I then \textstyle\sum\limits_ z_i = 0. And it can be shown that \textstyle\sum\limits_ r_i = \textstyle\sum\limits_ r_i + \textstyle\sum\limits_ r_i = 0 + \textstyle\sum\limits_ r_i = \textstyle\sum\limits_ r_i. If I = \N then the generalized series \textstyle\sum\limits_ r_i converges in \R if and only if \textstyle\sum\limits_^\infty r_i converges unconditionally (or equivalently, converges absolutely) in the usual sense. If a generalized series \textstyle\sum\limits_ r_i converges in \R then both \textstyle\sum\limits_ r_i and \textstyle\sum\limits_ r_i also converge to elements of \R and the set \left\ is necessarily
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
(that is, either finite or
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
); this remains true if \R is replaced with any normed space. It follows that in order for a generalized series \textstyle\sum\limits_ r_i to converge in \R or \Complex, it is necessary that all but at most countably many r_i will be equal to 0, which means that \textstyle\sum\limits_ r_i ~=~ \textstyle\sum\limits_ r_i is a sum of at most countably many non-zero terms. Said differently, if \left\ is uncountable then the generalized series \textstyle\sum\limits_ r_i does not converge. In summary, due to the nature of the real numbers and its topology, every generalized series of real numbers (indexed by an arbitrary set) that converges can be reduced to an ordinary absolutely convergent series of countably many real numbers. So in the context of measure theory, there is little benefit gained by considering uncountably many sets and generalized series. In particular, this is why the definition of " countably additive" is rarely extended from countably many sets F_1, F_2, \ldots\, in \mathcal (and the usual countable series \textstyle\sum\limits_^\infty \mu\left(F_i\right)) to arbitrarily many sets \left(F_i\right)_ (and the generalized series \textstyle\sum\limits_ \mu\left(F_i\right)).


Inner measures, outer measures, and other properties

A set function \mu is said to be/satisfies If a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...
\,+\, is defined, then a set function \mu is said to be


Topology related definitions

If \tau is a
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on \Omega then a set function \mu is said to be:


Relationships between set functions

If \mu and \nu are two set functions over \Omega, then:


Examples

Examples of set functions include: * The function d(A) = \lim_ \frac, assigning densities to sufficiently
well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or n ...
subsets A \subseteq \, is a set function. * A
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
assigns a probability to each set in a σ-algebra. Specifically, the probability of the
empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
is zero and the probability of the sample space is 1, with other sets given probabilities between 0 and 1. * A possibility measure assigns a number between zero and one to each set in the powerset of some given set. See possibility theory. * A is a set-valued
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
. See the article random compact set. The Jordan measure on \Reals^n is a set function defined on the set of all Jordan measurable subsets of \Reals^n; it sends a Jordan measurable set to its Jordan measure.


Lebesgue measure

The
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
on \Reals is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue \sigma-algebra.Kolmogorov and Fomin 1975 Its definition begins with the set \operatorname(\Reals) of all intervals of real numbers, which is a semialgebra on \Reals. The function that assigns to every interval I its \operatorname(I) is a finitely additive set function (explicitly, if I has endpoints a \leq b then \operatorname(I) = b - a). This set function can be extended to the Lebesgue outer measure on \Reals, which is the translation-invariant set function \lambda^ : \wp(\Reals) \to , \infty/math> that sends a subset E \subseteq \Reals to the infimum \lambda^(E) = \inf \left\. Lebesgue outer measure is not countably additive (and so is not a measure) although its restriction to the -algebra of all subsets M \subseteq \Reals that satisfy the Carathéodory criterion: \lambda^(M) = \lambda^(M \cap E) + \lambda^(M \cap E^c) \quad \text S \subseteq \Reals is a measure that called
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
. Vitali sets are examples of non-measurable sets of real numbers.


Infinite-dimensional space

As detailed in the article on infinite-dimensional Lebesgue measure, the only locally finite and translation-invariant
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. ...
on an infinite-dimensional separable normed space is the trivial measure. However, it is possible to define
Gaussian measure In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space \mathbb^n, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are na ...
s on infinite-dimensional
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s. The structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable
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 ...
.


Finitely additive translation-invariant set functions

The only translation-invariant measure on \Omega = \Reals with domain \wp(\Reals) that is finite on every compact subset of \Reals is the trivial set function \wp(\Reals) \to , \infty/math> that is identically equal to 0 (that is, it sends every S \subseteq \Reals to 0) However, if countable additivity is weakened to finite additivity then a non-trivial set function with these properties does exist and moreover, some are even valued in , 1 In fact, such non-trivial set functions will exist even if \Reals is replaced by any other abelian group G.


Extending set functions


Extending from semialgebras to algebras

Suppose that \mu is a set function on a semialgebra \mathcal over \Omega and let \operatorname(\mathcal) := \left\, which is the
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
on \Omega generated by \mathcal. The
archetypal The concept of an archetype ( ) appears in areas relating to behavior, History of psychology#Emergence of German experimental psychology, historical psychology, philosophy and literary analysis. An archetype can be any of the following: # a stat ...
example of a semialgebra that is not also an
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
is the family \mathcal_d := \ \cup \left\ on \Omega := \R^d where (a, b] := \ for all -\infty \leq a < b \leq \infty. Importantly, the two non-strict inequalities \,\leq\, in -\infty \leq a_i < b_i \leq \infty cannot be replaced with strict inequalities \,<\, since semialgebras must contain the whole underlying set \R^d; that is, \R^d \in \mathcal_d is a requirement of semialgebras (as is \varnothing \in \mathcal_d). If \mu is finitely additive then it has a unique extension to a set function \overline on \operatorname(\mathcal) defined by sending F_1 \sqcup \cdots \sqcup F_n \in \operatorname(\mathcal) (where \,\sqcup\, indicates that these F_i \in \mathcal are
pairwise disjoint In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
) to: \overline\left(F_1 \sqcup \cdots \sqcup F_n\right) := \mu\left(F_1\right) + \cdots + \mu\left(F_n\right). This extension \overline will also be finitely additive: for any pairwise disjoint A_1, \ldots, A_n \in \operatorname(\mathcal), \overline\left(A_1 \cup \cdots \cup A_n\right) = \overline\left(A_1\right) + \cdots + \overline\left(A_n\right). If in addition \mu is extended real-valued and monotone (which, in particular, will be the case if \mu is non-negative) then \overline will be monotone and finitely subadditive: for any A, A_1, \ldots, A_n \in \operatorname(\mathcal) such that A \subseteq A_1 \cup \cdots \cup A_n, \overline\left(A\right) \leq \overline\left(A_1\right) + \cdots + \overline\left(A_n\right).


Extending from rings to σ-algebras

If \mu : \mathcal \to , \infty/math> is a pre-measure on a ring of sets (such as an algebra of sets) \mathcal over \Omega then \mu has an extension to a measure \overline : \sigma(\mathcal) \to , \infty/math> on the σ-algebra \sigma(\mathcal) generated by \mathcal. If \mu is σ-finite then this extension is unique. To define this extension, first extend \mu to an outer measure \mu^* on 2^\Omega = \wp(\Omega) by \mu^*(T) = \inf \left\ and then restrict it to the set \mathcal_M of \mu^*-measurable sets (that is, Carathéodory-measurable sets), which is the set of all M \subseteq \Omega such that \mu^*(S) = \mu^*(S \cap M) + \mu^*(S \cap M^\mathrm) \quad \text S \subseteq \Omega. It is a \sigma-algebra and \mu^* is sigma-additive on it, by Caratheodory lemma.


Restricting outer measures

If \mu^* : \wp(\Omega) \to , \infty/math> is an outer measure on a set \Omega, where (by definition) the domain is necessarily 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 po ...
\wp(\Omega) of \Omega, then a subset M \subseteq \Omega is called or if it satisfies the following : \mu^*(S) = \mu^*(S \cap M) + \mu^*(S \cap M^\mathrm) \quad \text S \subseteq \Omega, where M^\mathrm := \Omega \setminus M is the complement of M. The family of all \mu^*–measurable subsets is a σ-algebra and the restriction of the outer measure \mu^* to this family is a measure.


See also

* * * * * * * * * * * * * *


Notes

Proofs


References

* * * A. N. Kolmogorov and S. V. Fomin (1975), ''Introductory Real Analysis'', Dover. * *


Further reading

*
Regular set function
a
Encyclopedia of Mathematics
{{Analysis in topological vector spaces Basic concepts in set theory Functions and mappings Measure theory Measures (measure theory)