Set function

{{Use American English|date = January 2019}} {{Short description|Function from sets to numbers}} In mathematics, especially measure theory, a '''set function''' is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line <math>\R \cup \{ \pm \infty \},</math> which consists of the real numbers <math>\R</math> and <math>\pm \infty.</math>

A set function generally aims to {{em|measure}} 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 <math>\mathcal{F}</math> is a family of sets over <math>\Omega</math> (meaning that <math>\mathcal{F} \subseteq \wp(\Omega)</math> where <math>\wp(\Omega)</math> denotes the powerset) then a {{em|set function on <math>\mathcal{F}</math>}} is a function <math>\mu</math> with domain <math>\mathcal{F}</math> and codomain <math>[-\infty, \infty]</math> or, sometimes, the codomain is instead some vector space, 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. {{Families of sets}}

In general, it is typically assumed that <math>\mu(E) + \mu(F)</math> is always well-defined for all <math>E, F \in \mathcal{F},</math> or equivalently, that <math>\mu</math> does not take on both <math>- \infty</math> and <math>+ \infty</math> 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 <math>\mu</math> is finitely additive: :{{em|{{visible anchor|Set difference formula}}}}: <math>\mu(F) - \mu(E) = \mu(F \setminus E) \text{ whenever } \mu(F) - \mu(E)</math> is defined with <math>E, F \in \mathcal{F}</math> satisfying <math>E \subseteq F</math> and <math>F \setminus E \in \mathcal{F}.</math>

'''Null sets'''

A set <math>F \in \mathcal{F}</math> is called a {{em|{{visible anchor|null set}}}} (with respect to <math>\mu</math>) or simply {{em|{{visible anchor|null}}}} if <math>\mu(F) = 0.</math> Whenever <math>\mu</math> is not identically equal to either <math>-\infty</math> or <math>+\infty</math> then it is typically also assumed that: <ul> <li>{{em|{{visible anchor|null empty set}}}}: <math>\mu(\varnothing) = 0</math> if <math>\varnothing \in \mathcal{F}.</math></li> </ul>

'''Variation and mass'''

The {{em|{{visible anchor|total variation of a set}}}} <math>S</math> is <math display=block>|\mu|(S) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \sup \{ |\mu(F)| : F \in \mathcal{F} \text{ and } F \subseteq S \}</math> where <math>|\,\cdot\,|</math> denotes the absolute value (or more generally, it denotes the norm or seminorm if <math>\mu</math> is vector-valued in a (semi)normed space). Assuming that <math>\cup \mathcal{F} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \textstyle\bigcup\limits_{F \in \mathcal{F}} F \in \mathcal{F},</math> then <math>|\mu|\left(\cup \mathcal{F}\right)</math> is called the {{em|{{visible anchor|total variation}}}} of <math>\mu</math> and <math>\mu\left(\cup \mathcal{F}\right)</math> is called the {{em|{{visible anchor|mass}}}} of <math>\mu.</math>

A set function is called {{em|{{visible anchor|finite}}}} if for every <math>F \in \mathcal{F},</math> the value <math>\mu(F)</math> is {{em|{{visible anchor|finite value|text=finite}}}} (which by definition means that <math>\mu(F) \neq \infty</math> and <math>\mu(F) \neq -\infty</math>; an {{em|{{visible anchor|infinite value}}}} is one that is equal to <math>\infty</math> or <math>- \infty</math>). Every finite set function must have a finite mass.

===Common properties of set functions===

A set function <math>\mu</math> on <math>\mathcal{F}</math> is said to be{{sfn|Durrett|2019|pp=1-37, 455-470}} <ul> <li>{{em|{{visible anchor|non-negative}}}} if it is valued in <math>[0, \infty].</math></li> <li>{{em|{{visible anchor|finitely additive}}}} if <math>\textstyle\sum\limits_{i=1}^n \mu\left(F_i\right) = \mu\left(\textstyle\bigcup\limits_{i=1}^n F_i\right)</math> for all pairwise disjoint finite sequences <math>F_1, \ldots, F_n \in \mathcal{F}</math> such that <math>\textstyle\bigcup\limits_{i=1}^n F_i \in \mathcal{F}.</math> * If <math>\mathcal{F}</math> is closed under binary unions then <math>\mu</math> is finitely additive if and only if <math>\mu(E \cup F) = \mu(E) + \mu(F)</math> for all disjoint pairs <math>E, F \in \mathcal{F}.</math> * If <math>\mu</math> is finitely additive and if <math>\varnothing \in \mathcal{F}</math> then taking <math>E := F := \varnothing</math> shows that <math>\mu(\varnothing) = \mu(\varnothing) + \mu(\varnothing)</math> which is only possible if <math>\mu(\varnothing) = 0</math> or <math>\mu(\varnothing) = \pm \infty,</math> where in the latter case, <math>\mu(E) = \mu(E \cup \varnothing) = \mu(E) + \mu(\varnothing) = \mu(E) + (\pm \infty) = \pm \infty</math> for every <math>E \in \mathcal{F}</math> (so only the case <math>\mu(\varnothing) = 0</math> is useful). </li> <li>{{em|{{visible anchor|countably additive}}}} or {{em|{{visible anchor|σ-additive}}}}{{sfn|Durrett|2019|pp=466-470}} if in addition to being finitely additive, for all pairwise disjoint sequences <math>F_1, F_2, \ldots\,</math> in <math>\mathcal{F}</math> such that <math>\textstyle\bigcup\limits_{i=1}^\infty F_i \in \mathcal{F},</math> all of the following hold: <ol type="a"> <li><math>\textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) = \mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right)</math> * The series on the left hand side is defined in the usual way as the limit <math>\textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) ~\stackrel{\scriptscriptstyle\text{def}}{=}~ {\displaystyle\lim_{n \to \infty}} \mu\left(F_1\right) + \cdots + \mu\left(F_n\right).</math> * As a consequence, if <math>\rho : \N \to \N</math> is any permutation/bijection then <math>\textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) = \textstyle\sum\limits_{i=1}^\infty \mu\left(F_{\rho(i)}\right);</math> this is because <math>\textstyle\bigcup\limits_{i=1}^\infty F_i = \textstyle\bigcup\limits_{i=1}^\infty F_{\rho(i)}</math> and applying this condition (a) twice guarantees that both <math>\textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) = \mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right)</math> and <math>\mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_{\rho(i)}\right) = \textstyle\sum\limits_{i=1}^\infty \mu\left(F_{\rho(i)}\right)</math> hold. By definition, a convergent series with this property is said to be unconditionally convergent. Stated in plain English, this means that rearranging/relabeling the sets <math>F_1, F_2, \ldots</math> to the new order <math>F_{\rho(1)}, F_{\rho(2)}, \ldots</math> does not affect the sum of their measures. This is desirable since just as the union <math>F ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \textstyle\bigcup\limits_{i \in \N} F_i</math> does not depend on the order of these sets, the same should be true of the sums <math>\mu(F) = \mu\left(F_1\right) + \mu\left(F_2\right) + \cdots</math> and <math>\mu(F) = \mu\left(F_{\rho(1)}\right) + \mu\left(F_{\rho(2)}\right) + \cdots\,.</math></li> <li>if <math>\mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right)</math> is not infinite then this series <math>\textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right)</math> must also converge absolutely, which by definition means that <math>\textstyle\sum\limits_{i=1}^\infty \left|\mu\left(F_i\right)\right|</math> must be finite. This is automatically true if <math>\mu</math> is non-negative (or even just valued in the extended real numbers). * As with any convergent series of real numbers, by the Riemann series theorem, the series <math>\textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right) = {\displaystyle\lim_{N \to \infty}} \mu\left(F_1\right) + \mu\left(F_2\right) + \cdots + \mu\left(F_N\right)</math> converges absolutely if and only if its sum does not depend on the order of its terms (a property known as unconditional convergence). Since unconditional convergence is guaranteed by (a) above, this condition is automatically true if <math>\mu</math> is valued in <math>[-\infty, \infty].</math></li> <li>if <math>\mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right) = \textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right)</math> is infinite then it is also required that the value of at least one of the series <math>\textstyle\sum\limits_{\stackrel{i \in \N}{\mu\left(F_i\right) > 0}} \mu\left(F_i\right) \; \text{ and } \; \textstyle\sum\limits_{\stackrel{i \in \N}{\mu\left(F_i\right) < 0}} \mu\left(F_i\right) \;</math> be finite (so that the sum of their values is well-defined). This is automatically true if <math>\mu</math> is non-negative.</li> </ol> </li> <li>a {{em|{{visible anchor|pre-measure}}}} if it is non-negative, countably additive (including finitely additive), and has a null empty set.</li> <li>a {{em|{{visible anchor|measure}}}} if it is a pre-measure whose domain is a σ-algebra. That is to say, a measure is a non-negative countably additive set function on a σ-algebra that has a null empty set.</li> <li>a {{em|{{visible anchor|probability measure}}}} if it is a measure that has a mass of <math>1.</math></li> <li>an {{em|{{visible anchor|outer measure}}}} if it is non-negative, countably subadditive, has a null empty set, and has the power set <math>\wp(\Omega)</math> as its domain. * Outer measures appear in the Carathéodory's extension theorem and they are often restricted to Carathéodory measurable subsets</li> <li>a {{em|{{visible anchor|signed measure}}}} if it is countably additive, has a null empty set, and <math>\mu</math> does not take on both <math>- \infty</math> and <math>+ \infty</math> as values.</li> <li>{{em|{{visible anchor|complete}}}} if every subset of every null set is null; explicitly, this means: whenever <math>F \in \mathcal{F} \text{ satisfies } \mu(F) = 0</math> and <math>N \subseteq F</math> is any subset of <math>F</math> then <math>N \in \mathcal{F}</math> and <math>\mu(N) = 0.</math> * Unlike many other properties, completeness places requirements on the set <math>\operatorname{domain} \mu = \mathcal{F}</math> (and not just on <math>\mu</math>'s values).</li> <li>{{em|{{visible anchor|{{sigma}}-finite}}}} if there exists a sequence <math>F_1, F_2, F_3, \ldots\,</math> in <math>\mathcal{F}</math> such that <math>\mu\left(F_i\right)</math> is finite for every index <math>i,</math> and also <math>\textstyle\bigcup\limits_{n=1}^\infty F_n = \textstyle\bigcup\limits_{F \in \mathcal{F}} F.</math></li> <li>{{em|{{visible anchor|decomposable}}}} if there exists a subfamily <math>\mathcal{P} \subseteq \mathcal{F}</math> of pairwise disjoint sets such that <math>\mu(P)</math> is finite for every <math>P \in \mathcal{P}</math> and also <math>\textstyle\bigcup\limits_{P \in \mathcal{P}} \, P = \textstyle\bigcup\limits_{F \in \mathcal{F}} F</math> (where <math>\mathcal{F} = \operatorname{domain} \mu</math>). * Every {{sigma}}-finite set function is decomposable although not conversely. For example, the counting measure on <math>\R</math> (whose domain is <math>\wp(\R)</math>) is decomposable but not {{sigma}}-finite.</li> <li>a {{em|{{visible anchor|vector measure}}}} if it is a countably additive set function <math>\mu : \mathcal{F} \to X</math> valued in a topological vector space <math>X</math> (such as a normed space) whose domain is a σ-algebra. * If <math>\mu</math> is valued in a normed space <math>(X, \|\cdot\|)</math> then it is countably additive if and only if for any pairwise disjoint sequence <math>F_1, F_2, \ldots\,</math> in <math>\mathcal{F},</math> <math>\lim_{n \to \infty} \left\|\mu\left(F_1\right) + \cdots + \mu\left(F_n\right) - \mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right)\right\| = 0.</math> If <math>\mu</math> is finitely additive and valued in a Banach space then it is countably additive if and only if for any pairwise disjoint sequence <math>F_1, F_2, \ldots\,</math> in <math>\mathcal{F},</math> <math>\lim_{n \to \infty} \left\|\mu\left(F_n \cup F_{n+1} \cup F_{n+2} \cup \cdots\right)\right\| = 0.</math></li> <li>a {{em|{{visible anchor|complex measure}}}} if it is a countably additive complex-valued set function <math>\mu : \mathcal{F} \to \Complex</math> whose domain is a σ-algebra. * By definition, a complex measure never takes <math>\pm \infty</math> as a value and so has a null empty set.</li> <li>a {{em|{{visible anchor|random measure}}}} if it is a measure-valued random element.</li> </ul>

'''Arbitrary sums'''

As described in this article's section on generalized series, for any family <math>\left(r_i\right)_{i \in I}</math> of real numbers indexed by an arbitrary indexing set <math>I,</math> it is possible to define their sum <math>\textstyle\sum\limits_{i \in I} r_i</math> as the limit of the net of finite partial sums <math>F \in \operatorname{FiniteSubsets}(I) \mapsto \textstyle\sum\limits_{i \in F} r_i</math> where the domain <math>\operatorname{FiniteSubsets}(I)</math> is directed by <math>\,\subseteq.\,</math> Whenever this net converges then its limit is denoted by the symbols <math>\textstyle\sum\limits_{i \in I} r_i</math> while if this net instead diverges to <math>\pm \infty</math> then this may be indicated by writing <math>\textstyle\sum\limits_{i \in I} r_i = \pm \infty.</math> Any sum over the empty set is defined to be zero; that is, if <math>I = \varnothing</math> then <math>\textstyle\sum\limits_{i \in \varnothing} r_i = 0</math> by definition.

For example, if <math>z_i = 0</math> for every <math>i \in I</math> then <math>\textstyle\sum\limits_{i \in I} z_i = 0.</math> And it can be shown that <math>\textstyle\sum\limits_{i \in I} r_i = \textstyle\sum\limits_{\stackrel{i \in I,}{r_i = 0}} r_i + \textstyle\sum\limits_{\stackrel{i \in I,}{r_i \neq 0}} r_i = 0 + \textstyle\sum\limits_{\stackrel{i \in I,}{r_i \neq 0}} r_i = \textstyle\sum\limits_{\stackrel{i \in I,}{r_i \neq 0}} r_i.</math> If <math>I = \N</math> then the generalized series <math>\textstyle\sum\limits_{i \in I} r_i</math> converges in <math>\R</math> if and only if <math>\textstyle\sum\limits_{i=1}^\infty r_i</math> converges unconditionally (or equivalently, converges absolutely) in the usual sense. If a generalized series <math>\textstyle\sum\limits_{i \in I} r_i</math> converges in <math>\R</math> then both <math>\textstyle\sum\limits_{\stackrel{i \in I}{r_i > 0}} r_i</math> and <math>\textstyle\sum\limits_{\stackrel{i \in I}{r_i < 0}} r_i</math> also converge to elements of <math>\R</math> and the set <math>\left\{i \in I : r_i \neq 0\right\}</math> is necessarily countable (that is, either finite or countably infinite); this remains true if <math>\R</math> is replaced with any normed space.<ref group=proof name=ProofCountablyManyNon0Terms /> It follows that in order for a generalized series <math>\textstyle\sum\limits_{i \in I} r_i</math> to converge in <math>\R</math> or <math>\Complex,</math> it is necessary that all but at most countably many <math>r_i</math> will be equal to <math>0,</math> which means that <math>\textstyle\sum\limits_{i \in I} r_i ~=~ \textstyle\sum\limits_{\stackrel{i \in I}{r_i \neq 0}} r_i</math> is a sum of at most countably many non-zero terms. Said differently, if <math>\left\{i \in I : r_i \neq 0\right\}</math> is uncountable then the generalized series <math>\textstyle\sum\limits_{i \in I} r_i</math> 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 <math>F_1, F_2, \ldots\,</math> in <math>\mathcal{F}</math> (and the usual countable series <math>\textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right)</math>) to arbitrarily many sets <math>\left(F_i\right)_{i \in I}</math> (and the generalized series <math>\textstyle\sum\limits_{i \in I} \mu\left(F_i\right)</math>).

===Inner measures, outer measures, and other properties===

A set function <math>\mu</math> is said to be/satisfies{{sfn|Durrett|2019|pp=1-37, 455-470}} <ul> <li>{{em|{{visible anchor|monotone}}}} if <math>\mu(E) \leq \mu(F)</math> whenever <math>E, F \in \mathcal{F}</math> satisfy <math>E \subseteq F.</math></li> <li>{{em|{{visible anchor|modular}}}} if it satisfies the following condition, known as {{em|modularity}}: <math>\mu(E \cup F) + \mu(E \cap F) = \mu(E) + \mu(F)</math> for all <math>E, F \in \mathcal{F}</math> such that <math>E \cup F, E \cap F \in \mathcal{F}.</math> * Every finitely additive function on a field of sets is modular. * In geometry, a set function valued in some abelian semigroup that possess this property is known as a {{em|valuation}}. This geometric definition of "valuation" should not be confused with the stronger non-equivalent measure theoretic definition of "valuation" that is given below.</li> <li>{{em|{{visible anchor|submodular}}}} if <math>\mu(E \cup F) + \mu(E \cap F) \leq \mu(E) + \mu(F)</math> for all <math>E, F \in \mathcal{F}</math> such that <math>E \cup F, E \cap F \in \mathcal{F}.</math></li> <li>{{em|{{visible anchor|finitely subadditive}}}} if <math>|\mu(F)| \leq \textstyle\sum\limits_{i=1}^n \left|\mu\left(F_i\right)\right|</math> for all finite sequences <math>F, F_1, \ldots, F_n \in \mathcal{F}</math> that satisfy <math>F \;\subseteq\; \textstyle\bigcup\limits_{i=1}^n F_i.</math></li> <li>{{em|{{visible anchor|countably subadditive}}}} or {{em|{{visible anchor|σ-subadditive}}}} if <math>|\mu(F)| \leq \textstyle\sum\limits_{i=1}^\infty \left|\mu\left(F_i\right)\right|</math> for all sequences <math>F, F_1, F_2, F_3, \ldots\,</math> in <math>\mathcal{F}</math> that satisfy <math>F \;\subseteq\; \textstyle\bigcup\limits_{i=1}^\infty F_i.</math> * If <math>\mathcal{F}</math> is closed under finite unions then this condition holds if and only if <math>|\mu(F \cup G)| \leq| \mu(F)| + |\mu(G)|</math> for all <math>F, G \in \mathcal{F}.</math> If <math>\mu</math> is non-negative then the absolute values may be removed. * If <math>\mu</math> is a measure then this condition holds if and only if <math>\mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right) \leq \textstyle\sum\limits_{i=1}^\infty \mu\left(F_i\right)</math> for all <math>F_1, F_2, F_3, \ldots\,</math> in <math>\mathcal{F}.</math>{{sfn|Royden|Fitzpatrick|2010|p=30}} If <math>\mu</math> is a probability measure then this inequality is Boole's inequality. * If <math>\mu</math> is countably subadditive and <math>\varnothing \in \mathcal{F}</math> with <math>\mu(\varnothing) = 0</math> then <math>\mu</math> is finitely subadditive.</li> <li>{{em|{{visible anchor|superadditive}}}} if <math>\mu(E) + \mu(F) \leq \mu(E \cup F)</math> whenever <math>E, F \in \mathcal{F}</math> are disjoint with <math>E \cup F \in \mathcal{F}.</math></li> <li>{{em|{{visible anchor|continuous from above}}}} if <math>\lim_{n \to \infty} \mu\left(F_i\right) = \mu\left(\textstyle\bigcap\limits_{i=1}^\infty F_i\right)</math> for all {{em|non-increasing sequences}} of sets <math>F_1 \supseteq F_2 \supseteq F_3 \cdots\,</math> in <math>\mathcal{F}</math> such that <math>\textstyle\bigcap\limits_{i=1}^\infty F_i \in \mathcal{F}</math> with <math>\mu\left(\textstyle\bigcap\limits_{i=1}^\infty F_i\right)</math> and all <math>\mu\left(F_i\right)</math> finite. * Lebesgue measure <math>\lambda</math> is continuous from above but it would not be if the assumption that all <math>\mu\left(F_i\right)</math> are eventually finite was omitted from the definition, as this example shows: For every integer <math>i,</math> let <math>F_i</math> be the open interval <math>(i, \infty)</math> so that <math>\lim_{n \to \infty} \lambda\left(F_i\right) = \lim_{n \to \infty} \infty = \infty \neq 0 = \lambda(\varnothing) = \lambda\left(\textstyle\bigcap\limits_{i=1}^\infty F_i\right)</math> where <math>\textstyle\bigcap\limits_{i=1}^\infty F_i = \varnothing.</math></li> <li>{{em|{{visible anchor|continuous from below}}}} if <math>\lim_{n \to \infty} \mu\left(F_i\right) = \mu\left(\textstyle\bigcup\limits_{i=1}^\infty F_i\right)</math> for all {{em|non-decreasing sequences}} of sets <math>F_1 \subseteq F_2 \subseteq F_3 \cdots\,</math> in <math>\mathcal{F}</math> such that <math>\textstyle\bigcup\limits_{i=1}^\infty F_i \in \mathcal{F}.</math></li> <li>{{em|{{visible anchor|infinity is approached from below}}}} if whenever <math>F \in \mathcal{F}</math> satisfies <math>\mu(F) = \infty</math> then for every real <math>r > 0,</math> there exists some <math>F_r \in \mathcal{F}</math> such that <math>F_r \subseteq F</math> and <math>r \leq \mu\left(F_r\right) < \infty.</math></li> <li>an {{em|outer measure}} if <math>\mu</math> is non-negative, countably subadditive, has a null empty set, and has the power set <math>\wp(\Omega)</math> as its domain.</li> <li>an {{em|{{visible anchor|inner measure}}}} if <math>\mu</math> is non-negative, superadditive, continuous from above, has a null empty set, has the power set <math>\wp(\Omega)</math> as its domain, and <math>+\infty</math> is approached from below.</li> <li>{{em|atomic}} if every measurable set of positive measure contains an atom.</li> </ul>

If a binary operation <math>\,+\,</math> is defined, then a set function <math>\mu</math> is said to be <ul> <li>{{em|{{visible anchor|translation invariant}}}} if <math>\mu(\omega + F) = \mu(F)</math> for all <math>\omega \in \Omega</math> and <math>F \in \mathcal{F}</math> such that <math>\omega + F \in \mathcal{F}.</math></li> </ul>

===Topology related definitions===

If <math>\tau</math> is a topology on <math>\Omega</math> then a set function <math>\mu</math> is said to be: <ul> <li>a {{em|{{visible anchor|Borel measure}}}} if it is a measure defined on the σ-algebra of all Borel sets, which is the smallest σ-algebra containing all open subsets (that is, containing <math>\tau</math>).</li> <li>a {{em|{{visible anchor|Baire measure}}}} if it is a measure defined on the σ-algebra of all Baire sets.</li> <li>{{em|{{visible anchor|locally finite}}}} if for every point <math>\omega \in \Omega</math> there exists some neighborhood <math>U \in \mathcal{F} \cap \tau</math> of this point such that <math>\mu(U)</math> is finite. * If <math>\mu</math> is a finitely additive, monotone, and locally finite then <math>\mu(K)</math> is necessarily finite for every compact measurable subset <math>K.</math></li> <li>{{em|{{visible anchor|<math>\tau</math>-additive}}}} if <math>\mu\left({\textstyle\bigcup} \, \mathcal{D}\right) = \sup_{D \in \mathcal{D}} \mu(D)</math> whenever <math>\mathcal{D} \subseteq \tau \cap \mathcal{F}</math> is directed with respect to <math>\,\subseteq\,</math> and satisfies <math>{\textstyle\bigcup} \, \mathcal{D} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \textstyle\bigcup\limits_{D \in \mathcal{D}} D \in \mathcal{F}.</math> * <math>\mathcal{D}</math> is directed with respect to <math>\,\subseteq\,</math> if and only if it is not empty and for all <math>A, B \in \mathcal{D}</math> there exists some <math>C \in \mathcal{D}</math> such that <math>A \subseteq C</math> and <math>B \subseteq C.</math></li> <li>{{em|{{visible anchor|inner regular}}}} or {{em|{{visible anchor|tight}}}} if for every <math>F \in \mathcal{F},</math> <math>\mu(F) = \sup \{\mu(K) : F \supseteq K \text{ with } K \in \mathcal{F} \text{ a compact subset of } (\Omega, \tau)\}.</math></li> <li>{{em|{{visible anchor|outer regular}}}} if for every <math>F \in \mathcal{F},</math> <math>\mu(F) = \inf \{\mu(U) : F \subseteq U \text{ and } U \in \mathcal{F} \cap \tau\}.</math></li> <li>{{em|{{visible anchor|regular}}}} if it is both inner regular and outer regular.</li> <li>a {{em|{{visible anchor|Borel regular measure}}}} if it is a Borel measure that is also {{em|regular}}.</li> <li>a {{em|{{visible anchor|Radon measure}}}} if it is a regular and locally finite measure.</li> <li>{{em|{{visible anchor|strictly positive}}}} if every non-empty open subset has (strictly) positive measure.</li> <li>a {{em|{{visible anchor|valuation}}}} if it is non-negative, monotone, modular, has a null empty set, and has domain <math>\tau.</math></li> </ul>

===Relationships between set functions=== {{See also|Radon–Nikodym theorem|Lebesgue's decomposition theorem}}

If <math>\mu</math> and <math>\nu</math> are two set functions over <math>\Omega,</math> then: <ul> <li><math>\mu</math> is said to be {{em|{{visible anchor|absolutely continuous}} with respect to <math>\nu</math>}} or {{em|dominated by <math>\nu</math>}}, written <math>\mu \ll \nu,</math> if for every set <math>F</math> that belongs to the domain of both <math>\mu</math> and <math>\nu,</math> if <math>\nu(F) = 0</math> then <math>\mu(F) = 0.</math> * If <math>\mu</math> and <math>\nu</math> are <math>\sigma</math>-finite measures on the same measurable space and if <math>\mu \ll \nu,</math> then the Radon–Nikodym derivative <math>\frac{d \mu}{d \nu}</math> exists and for every measurable <math>F,</math> <math display=block>\mu(F) = \int_F \frac{d \mu}{d \nu} d \nu.</math></li> * <math>\mu</math> and <math>\nu</math> are called {{em|{{visible anchor|equivalent}}}} if each one is absolutely continuous with respect to the other. <math>\mu</math> is called a {{em|{{visible anchor|supporting measure}}}} of a measure <math>\nu</math> if <math>\mu</math> is <math>\sigma</math>-finite and they are equivalent.<ref>{{cite book |last1=Kallenberg |first1=Olav |author-link1=Olav Kallenberg |year=2017 |title=Random Measures, Theory and Applications|series=Probability Theory and Stochastic Modelling |volume=77 |location= Switzerland |publisher=Springer |doi= 10.1007/978-3-319-41598-7|isbn=978-3-319-41596-3|page=21}}</ref> <li><math>\mu</math> and <math>\nu</math> are {{em|{{visible anchor|singular}}}}, written <math>\mu \perp \nu,</math> if there exist disjoint sets <math>M</math> and <math>N</math> in the domains of <math>\mu</math> and <math>\nu</math> such that <math>M \cup N = \Omega,</math> <math>\mu(F) = 0</math> for all <math>F \subseteq M</math> in the domain of <math>\mu,</math> and <math>\nu(F) = 0</math> for all <math>F \subseteq N</math> in the domain of <math>\nu.</math></li> </ul>

==Examples==

Examples of set functions include: * The function <math display=block>d(A) = \lim_{n \to \infty} \frac{|A \cap \{1, \ldots, n\}|}{n},</math> assigning densities to sufficiently well-behaved subsets <math>A \subseteq \{1, 2, 3, \ldots\},</math> is a set function. * A probability measure assigns a probability to each set in a σ-algebra. Specifically, the probability of the empty set is zero and the probability of the sample space is <math>1,</math> with other sets given probabilities between <math>0</math> and <math>1.</math> * A possibility measure assigns a number between zero and one to each set in the powerset of some given set. See possibility theory. * A {{em|random set}} is a set-valued random variable. See the article random compact set.

The Jordan measure on <math>\Reals^n</math> is a set function defined on the set of all Jordan measurable subsets of <math>\Reals^n;</math> it sends a Jordan measurable set to its Jordan measure.

===Lebesgue measure===

The Lebesgue measure on <math>\Reals</math> is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue <math>\sigma</math>-algebra.<ref>Kolmogorov and Fomin 1975</ref>

Its definition begins with the set <math>\operatorname{Intervals}(\Reals)</math> of all intervals of real numbers, which is a semialgebra on <math>\Reals.</math> The function that assigns to every interval <math>I</math> its <math>\operatorname{length}(I)</math> is a finitely additive set function (explicitly, if <math>I</math> has endpoints <math>a \leq b</math> then <math>\operatorname{length}(I) = b - a</math>). This set function can be extended to the Lebesgue outer measure on <math>\Reals,</math> which is the translation-invariant set function <math>\lambda^{\!*\!} : \wp(\Reals) \to [0, \infty]</math> that sends a subset <math>E \subseteq \Reals</math> to the infimum <math display=block>\lambda^{\!*\!}(E) = \inf \left\{\sum_{k=1}^\infty \operatorname{length}(I_k) : {(I_k)_{k \in \N}} \text{ is a sequence of open intervals with } E \subseteq \bigcup_{k=1}^\infty I_k\right\}.</math> Lebesgue outer measure is not countably additive (and so is not a measure) although its restriction to the {{sigma}}-algebra of all subsets <math>M \subseteq \Reals</math> that satisfy the Carathéodory criterion: <math display=block>\lambda^{\!*\!}(M) = \lambda^{\!*\!}(M \cap E) + \lambda^{\!*\!}(M \cap E^c) \quad \text{ for every } S \subseteq \Reals</math> is a measure that called Lebesgue measure. Vitali sets are examples of non-measurable sets of real numbers.

====Infinite-dimensional space====

{{See also|Gaussian measure#Infinite-dimensional spaces|Abstract Wiener space|Feldman–Hájek theorem|Radonifying function}}

As detailed in the article on infinite-dimensional Lebesgue measure, the only locally finite and translation-invariant Borel measure on an infinite-dimensional separable normed space is the trivial measure. However, it is possible to define Gaussian measures on infinite-dimensional topological vector spaces. 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.

===Finitely additive translation-invariant set functions===

The only translation-invariant measure on <math>\Omega = \Reals</math> with domain <math>\wp(\Reals)</math> that is finite on every compact subset of <math>\Reals</math> is the trivial set function <math>\wp(\Reals) \to [0, \infty]</math> that is identically equal to <math>0</math> (that is, it sends every <math>S \subseteq \Reals</math> to <math>0</math>){{sfn|Rudin|1991|p=139}} 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 <math>[0, 1].</math> In fact, such non-trivial set functions will exist even if <math>\Reals</math> is replaced by any other abelian group <math>G.</math>{{sfn|Rudin|1991|pp=139-140}}

{{Math theorem | name = Theorem{{sfn|Rudin|1991|pp=141-142}} | math_statement = If <math>(G, +)</math> is any abelian group then there exists a finitely additive and translation-invariant<ref group=note name=GroupTranslation>The function <math>\mu</math> being translation-invariant means that <math>\mu(S) = \mu(g + S)</math> for every <math>g \in G</math> and every subset <math>S \subseteq G.</math></ref> set function <math>\mu : \wp(G) \to [0, 1]</math> of mass <math>\mu(G) = 1.</math> }}

==Extending set functions== {{See also|Carathéodory's extension theorem}}

===Extending from semialgebras to algebras===

Suppose that <math>\mu</math> is a set function on a semialgebra <math>\mathcal{F}</math> over <math>\Omega</math> and let <math display=block>\operatorname{algebra}(\mathcal{F}) := \left\{ F_1 \sqcup \cdots \sqcup F_n : n \in \N \text{ and } F_1, \ldots, F_n \in \mathcal{F} \text{ are pairwise disjoint } \right\},</math> which is the algebra on <math>\Omega</math> generated by <math>\mathcal{F}.</math> The archetypal example of a semialgebra that is not also an algebra is the family <math display=block>\mathcal{S}_d := \{ \varnothing \} \cup \left\{ \left(a_1, b_1\right] \times \cdots \times \left(a_1, b_1\right] ~:~ -\infty \leq a_i < b_i \leq \infty \text{ for all } i = 1, \ldots, d \right\}</math> on <math>\Omega := \R^d</math> where <math>(a, b] := \{ x \in \R : a < x \leq b \}</math> for all <math>-\infty \leq a < b \leq \infty.</math>{{sfn|Durrett|2019|pp=1-9}} Importantly, the two non-strict inequalities <math>\,\leq\,</math> in <math>-\infty \leq a_i < b_i \leq \infty</math> cannot be replaced with strict inequalities <math>\,<\,</math> since semialgebras must contain the whole underlying set <math>\R^d;</math> that is, <math>\R^d \in \mathcal{S}_d</math> is a requirement of semialgebras (as is <math>\varnothing \in \mathcal{S}_d</math>).

If <math>\mu</math> is finitely additive then it has a unique extension to a set function <math>\overline{\mu}</math> on <math>\operatorname{algebra}(\mathcal{F})</math> defined by sending <math>F_1 \sqcup \cdots \sqcup F_n \in \operatorname{algebra}(\mathcal{F})</math> (where <math>\,\sqcup\,</math> indicates that these <math>F_i \in \mathcal{F}</math> are pairwise disjoint) to:{{sfn|Durrett|2019|pp=1-9}} <math display=block>\overline{\mu}\left(F_1 \sqcup \cdots \sqcup F_n\right) := \mu\left(F_1\right) + \cdots + \mu\left(F_n\right).</math> This extension <math>\overline{\mu}</math> will also be finitely additive: for any pairwise disjoint <math>A_1, \ldots, A_n \in \operatorname{algebra}(\mathcal{F}),</math> {{sfn|Durrett|2019|pp=1-9}} <math display=block>\overline{\mu}\left(A_1 \cup \cdots \cup A_n\right) = \overline{\mu}\left(A_1\right) + \cdots + \overline{\mu}\left(A_n\right).</math>

If in addition <math>\mu</math> is extended real-valued and monotone (which, in particular, will be the case if <math>\mu</math> is non-negative) then <math>\overline{\mu}</math> will be monotone and finitely subadditive: for any <math>A, A_1, \ldots, A_n \in \operatorname{algebra}(\mathcal{F})</math> such that <math>A \subseteq A_1 \cup \cdots \cup A_n,</math>{{sfn|Durrett|2019|pp=1-9}} <math display=block>\overline{\mu}\left(A\right) \leq \overline{\mu}\left(A_1\right) + \cdots + \overline{\mu}\left(A_n\right).</math>

===Extending from rings to σ-algebras=== {{See also|Pre-measure|Hahn–Kolmogorov theorem}}

If <math>\mu : \mathcal{F} \to [0, \infty]</math> is a pre-measure on a ring of sets (such as an algebra of sets) <math>\mathcal{F}</math> over <math>\Omega</math> then <math>\mu</math> has an extension to a measure <math>\overline{\mu} : \sigma(\mathcal{F}) \to [0, \infty]</math> on the σ-algebra <math>\sigma(\mathcal{F})</math> generated by <math>\mathcal{F}.</math> If <math>\mu</math> is σ-finite then this extension is unique.

To define this extension, first extend <math>\mu</math> to an outer measure <math>\mu^*</math> on <math>2^\Omega = \wp(\Omega)</math> by <math display=block>\mu^*(T) = \inf \left\{\sum_n \mu\left(S_n\right) : T \subseteq \cup_n S_n \text{ with } S_1, S_2, \ldots \in \mathcal{F}\right\}</math> and then restrict it to the set <math>\mathcal{F}_M</math> of <math>\mu^*</math>-measurable sets (that is, Carathéodory-measurable sets), which is the set of all <math>M \subseteq \Omega</math> such that <math display=block>\mu^*(S) = \mu^*(S \cap M) + \mu^*(S \cap M^\mathrm{c}) \quad \text{ for every subset } S \subseteq \Omega.</math> It is a <math>\sigma</math>-algebra and <math>\mu^*</math> is sigma-additive on it, by Caratheodory lemma.

===Restricting outer measures=== {{See also|Outer measure#Measurability of sets relative to an outer measure}}

If <math>\mu^* : \wp(\Omega) \to [0, \infty]</math> is an outer measure on a set <math>\Omega,</math> where (by definition) the domain is necessarily the power set <math>\wp(\Omega)</math> of <math>\Omega,</math> then a subset <math>M \subseteq \Omega</math> is called '''{{em|<math>\mu^*</math>–measurable}}''' or '''{{em|Carathéodory-measurable}}''' if it satisfies the following {{em|Carathéodory's criterion}}: <math display=block>\mu^*(S) = \mu^*(S \cap M) + \mu^*(S \cap M^\mathrm{c}) \quad \text{ for every subset } S \subseteq \Omega,</math> where <math>M^\mathrm{c} := \Omega \setminus M</math> is the complement of <math>M.</math>

The family of all <math>\mu^*</math>–measurable subsets is a σ-algebra and the restriction of the outer measure <math>\mu^*</math> to this family is a measure.

==See also==

* {{annotated link|Absolute continuity (measure theory)}} * {{annotated link|Boolean ring}} * {{annotated link|Cylinder set measure}} * {{annotated link|Field of sets}} * {{annotated link|Hadwiger's theorem}} * {{annotated link|Hahn decomposition theorem}} * {{annotated link|Invariant measure}} * {{annotated link|Lebesgue's decomposition theorem}} * {{annotated link|Positive and negative sets}} * {{annotated link|Radon–Nikodym theorem}} * {{annotated link|Riesz–Markov–Kakutani representation theorem}} * {{annotated link|Ring of sets}} * {{annotated link|σ-algebra}} * {{annotated link|Vitali–Hahn–Saks theorem}}

==Notes==

'''Proofs'''

{{reflist|group=proof|refs= <ref name=ProofCountablyManyNon0Terms>Suppose the net <math display=inline>\textstyle\sum\limits_{i \in I} r_i ~\stackrel{\scriptscriptstyle\text{def}}{=}~ {\textstyle\lim\limits_{A \in \operatorname{Finite}(I)}} \ \textstyle\sum\limits_{i \in A} r_i = \lim \left\{\textstyle\sum\limits_{i\in A} r_i \,: A \subseteq I, A \text{ finite }\right\}</math> converges to some point in a metrizable topological vector space <math>X</math> (such as <math>\Reals,</math> <math>\Complex,</math> or a normed space), where recall that this net's domain is the directed set <math>(\operatorname{Finite}(I), \subseteq).</math> Like every convergent net, this convergent net of partial sums <math>A \mapsto \textstyle\sum\limits_{i \in A} r_i</math> is a {{em|Cauchy net}}, which for this particular net means (by definition) that for every neighborhood <math>W</math> of the origin in <math>X,</math> there exists a finite subset <math>A_0</math> of <math>I</math> such that <math display=inline>\textstyle\sum\limits_{i \in B} r_i - \textstyle\sum\limits_{i \in C} r_i \in W</math> for all finite supersets <math>B, C \supseteq A_0;</math> this implies that <math>r_i \in W</math> for every <math>i \in I \setminus A_0</math> (by taking <math>B := A_0 \cup \{i\}</math> and <math>C := A_0</math>). Since <math>X</math> is metrizable, it has a countable neighborhood basis <math>U_1, U_2, \ldots</math> at the origin, whose intersection is necessarily <math>U_1 \cap U_2 \cap \cdots = \{0\}</math> (since <math>X</math> is a Hausdorff TVS). For every positive integer <math>n \in \N,</math> pick a finite subset <math>A_n \subseteq I</math> such that <math>r_i \in U_n</math> for every <math>i \in I \setminus A_n.</math> If <math>i</math> belongs to <math>(I \setminus A_1) \cap (I \setminus A_2) \cap \cdots = I \setminus \left(A_1 \cup A_2 \cup \cdots\right)</math> then <math>r_i</math> belongs to <math>U_1 \cap U_2 \cap \cdots = \{0\}.</math> Thus <math>r_i = 0</math> for every index <math>i \in I</math> that does not belong to the countable set <math>A_1 \cup A_2 \cup \cdots.</math> <math>\blacksquare</math></ref> }}

==References== {{sfn whitelist|CITEREFDurrett2019|CITEREFRoydenFitzpatrick2010}} * {{Durrett Probability Theory and Examples 5th Edition}}  * {{Kolmogorov Fomin Elements of the Theory of Functions and Functional Analysis}}  * A. N. Kolmogorov and S. V. Fomin (1975), ''Introductory Real Analysis'', Dover. {{isbn|0-486-61226-0}} * {{Royden Fitzpatrick Real Analysis 4th 2010}}  * {{Rudin Walter Functional Analysis|edition=2}}

==Further reading==

* {{springer|title=Set function|id=S/s084730|last=Sobolev|first=V.I.}} * [http://www.encyclopediaofmath.org/index.php/Regular_set_function Regular set function] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

Category:Basic concepts in set theory Category:Functions and mappings Category:Measure theory Category:Measures (measure theory)