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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 mass and probability of events. These seemingly distinct concepts have many simi ...
, a set function 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-oriente ...
whose
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
is a
family Family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its members and of society. Ideall ...
of
subset In mathematics, set ''A'' is a subset of a set ''B'' if all 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 are unequal, then ''A'' is a proper subset o ...
s of some given set and that (usually) takes its values in the
extended real number line In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra o ...
\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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
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 collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set fa ...
over \Omega (meaning that \mathcal \subseteq \wp(\Omega) where \wp(\Omega) denotes the
powerset 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 p ...
) then a is a function \mu with
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
\mathcal and
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
\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 whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
, as with
vector measure In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only. Definitions a ...
s, complex measures, and
projection-valued measure In mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are ...
s. The domain is 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 In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A fun ...
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 In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivit ...
: :: \mu(F) - \mu(E) = \mu(F \setminus E) \text \mu(F) - \mu(E) is defined with E, F \in \mathcal \text E \subseteq F \text 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) := \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 is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
(or more generally, it denotes the
norm Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
or
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk a ...
if \mu is vector-valued in a (
semi SEMI is an industry association comprising companies involved in the electronics design and manufacturing supply chain. They provide equipment, materials and services for the manufacture of semiconductors, photovoltaic panels, LED and flat p ...
)
normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
). Assuming that \cup \mathcal := \bigcup_ 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 be 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 property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementa ...
.


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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
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 Director may refer to: Literature * ''Director'' (magazine), a British magazine * ''The Director'' (novel), a 1971 novel by Henry Denker * ''The Director'' (play), a 2000 play by Nancy Hasty Music * Director (band), an Irish rock band * ''Di ...
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 = \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 mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is sai ...
) 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 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 numbers ...
(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 number ...
); this remains true if \R is replaced with any
normed space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
. 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 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 In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivit ...
" 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, an internal binary op ...
\,+\, is defined, then a set function \mu is said to be


Topology related definitions

If \tau is a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
on \Omega then a set function \mu is said to be:


Relationships between set functions

If \mu \text \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. T ...
subsets A \subseteq \, is a set function. * 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 ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides w ...
is a set function that assigns a non-negative real number to any set of real numbers, that is in Lebesgue \sigma-algebra.Kolmogorov and Fomin 1975 * A
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
assigns a probability to each set in a
σ-algebra In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countable ...
. Specifically, the probability of the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
is zero and the probability of the
sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually de ...
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 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 p ...
of some given set. See
possibility theory Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessa ...
. * A is a set-valued
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
. See the article random compact set.


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 one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
on \Omega generated by \mathcal. The
archetypal The concept of an archetype (; ) appears in areas relating to behavior, historical psychology, and literary analysis. An archetype can be any of the following: # a statement, pattern of behavior, prototype, "first" form, or a main model that ot ...
example of a semialgebra that is not also an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
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 In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, \mu(A \cup B) = \mu(A) + \mu(B). If this additivit ...
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 mathematics, 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,'' while and are not disjoint. A ...
) 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 In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or i ...
) 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 In mathematics, a pre-measure is a set function that is, in some sense, a precursor to a ''bona fide'' measure on a given space. Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure. Def ...
on a
ring of sets In mathematics, there are two different notions of a ring of sets, both referring to certain families of sets. In order theory, a nonempty family of sets \mathcal is called a ring (of sets) if it is closed under union and intersection.. That i ...
(such as an
algebra of sets In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the re ...
) \mathcal over \Omega then \mu has an extension to a measure \overline : \sigma(\mathcal) \to , \infty/math> on the
σ-algebra In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countable ...
\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 In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer me ...
\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 In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer me ...
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 p ...
\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 A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
of M. The family of all \mu^*–measurable subsets is a
σ-algebra In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countable ...
and the restriction of the outer measure \mu^* to this family is a measure.


See also

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


Notes


References

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


Further reading

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