set function

If $\backslash mathcal\{F\}$ is a family of sets over $\backslash Omega$ (meaning that $\backslash mathcal\{F\}\; \backslash subseteq\; \backslash wp(\backslash Omega)$ where $\backslash wp(\backslash Omega)$ denotes the powerset) then a {{em|set function on $\backslash mathcal\{F\}$}} is a function $\backslash mu$ with domain $\backslash mathcal\{F\}$ and codomain $[-\backslash infty,\; \backslash infty]$ 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 $\backslash mu(E)\; +\; \backslash mu(F)$ is always well-defined for all $E,\; F\; \backslash in\; \backslash mathcal\{F\},$ or equivalently, that $\backslash mu$ does not take on both $-\; \backslash infty$ and $+\; \backslash infty$ 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 $\backslash mu$ is finitely additive:

:{{em|{{visible anchor|Set difference formula}}}}: $\backslash mu(F)\; -\; \backslash mu(E)\; =\; \backslash mu(F\; \backslash setminus\; E)\; \backslash text\{\; whenever\; \}\; \backslash mu(F)\; -\; \backslash mu(E)$ is defined with $E,\; F\; \backslash in\; \backslash mathcal\{F\}$ satisfying $E\; \backslash subseteq\; F$ and $F\; \backslash setminus\; E\; \backslash in\; \backslash mathcal\{F\}.$

Null sets

A set $F\; \backslash in\; \backslash mathcal\{F\}$ is called a {{em|{{visible anchor|null set}}}} (with respect to $\backslash mu$) or simply {{em|{{visible anchor|null}}}} if $\backslash mu(F)\; =\; 0.$

Whenever $\backslash mu$ is not identically equal to either $-\backslash infty$ or $+\backslash infty$ then it is typically also assumed that:

• {{em|{{visible anchor|null empty set}}}}: $\backslash mu(\backslash varnothing)\; =\; 0$ if $\backslash varnothing\; \backslash in\; \backslash mathcal\{F\}.$

Variation and mass

The Total variation (measure theory) $S$ is

$$|\backslash mu|(S)\; ~\backslash stackrel\{\backslash scriptscriptstyle\backslash text\{def\}\}\{=\}~\; \backslash sup\; \backslash \{\; |\backslash mu(F)|\; :\; F\; \backslash in\; \backslash mathcal\{F\}\; \backslash text\{\; and\; \}\; F\; \backslash subseteq\; S\; \backslash \}$$

where $|\backslash ,\backslash cdot\backslash ,|$ denotes the absolute value (or more generally, it denotes the norm or seminorm if $\backslash mu$ is vector-valued in a (semi)normed space).

Assuming that $\backslash cup\; \backslash mathcal\{F\}\; ~\backslash stackrel\{\backslash scriptscriptstyle\backslash text\{def\}\}\{=\}~\; \backslash textstyle\backslash bigcup\backslash limits\_\{F\; \backslash in\; \backslash mathcal\{F\}\}\; F\; \backslash in\; \backslash mathcal\{F\},$ then $|\backslash mu|\backslash left(\backslash cup\; \backslash mathcal\{F\}\backslash right)$ is called the {{em|{{visible anchor|total variation}}}} of $\backslash mu$ and $\backslash mu\backslash left(\backslash cup\; \backslash mathcal\{F\}\backslash right)$ is called the {{em|{{visible anchor|mass}}}} of $\backslash mu.$

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

Every finite set function must have a finite mass.

A set function $\backslash mu$ on $\backslash mathcal\{F\}$ is said to be{{sfn|Durrett|2019|pp=1-37, 455-470}}

• {{em|{{visible anchor|non-negative}}}} if it is valued in $[0,\; \backslash infty].$
• Finitely additive set function if $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^n\; \backslash mu\backslash left(F\_i\backslash right)\; =\; \backslash mu\backslash left(\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^n\; F\_i\backslash right)$ for all pairwise disjoint finite sequences $F\_1,\; \backslash ldots,\; F\_n\; \backslash in\; \backslash mathcal\{F\}$ such that $\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^n\; F\_i\; \backslash in\; \backslash mathcal\{F\}.$
  • If $\backslash mathcal\{F\}$ is closed under binary unions then $\backslash mu$ is finitely additive if and only if $\backslash mu(E\; \backslash cup\; F)\; =\; \backslash mu(E)\; +\; \backslash mu(F)$ for all disjoint pairs $E,\; F\; \backslash in\; \backslash mathcal\{F\}.$
  • If $\backslash mu$ is finitely additive and if $\backslash varnothing\; \backslash in\; \backslash mathcal\{F\}$ then taking $E\; :=\; F\; :=\; \backslash varnothing$ shows that $\backslash mu(\backslash varnothing)\; =\; \backslash mu(\backslash varnothing)\; +\; \backslash mu(\backslash varnothing)$ which is only possible if $\backslash mu(\backslash varnothing)\; =\; 0$ or $\backslash mu(\backslash varnothing)\; =\; \backslash pm\; \backslash infty,$ where in the latter case, $\backslash mu(E)\; =\; \backslash mu(E\; \backslash cup\; \backslash varnothing)\; =\; \backslash mu(E)\; +\; \backslash mu(\backslash varnothing)\; =\; \backslash mu(E)\; +\; (\backslash pm\; \backslash infty)\; =\; \backslash pm\; \backslash infty$ for every $E\; \backslash in\; \backslash mathcal\{F\}$ (so only the case $\backslash mu(\backslash varnothing)\; =\; 0$ is useful).

• Sigma-additive set function or Sigma-additive set function{{sfn|Durrett|2019|pp=466-470}} if in addition to being finitely additive, for all pairwise disjoint sequences $F\_1,\; F\_2,\; \backslash ldots\backslash ,$ in $\backslash mathcal\{F\}$ such that $\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_i\; \backslash in\; \backslash mathcal\{F\},$ all of the following hold:

  1. $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash mu\backslash left(F\_i\backslash right)\; =\; \backslash mu\backslash left(\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_i\backslash right)$
     • The series on the left hand side is defined in the usual way as the limit $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash mu\backslash left(F\_i\backslash right)\; ~\backslash stackrel\{\backslash scriptscriptstyle\backslash text\{def\}\}\{=\}~\; \{\backslash displaystyle\backslash lim\_\{n\; \backslash to\; \backslash infty\}\}\; \backslash mu\backslash left(F\_1\backslash right)\; +\; \backslash cdots\; +\; \backslash mu\backslash left(F\_n\backslash right).$
     • As a consequence, if $\backslash rho\; :\; \backslash N\; \backslash to\; \backslash N$ is any permutation/bijection then $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash mu\backslash left(F\_i\backslash right)\; =\; \backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash mu\backslash left(F\_\{\backslash rho(i)\}\backslash right);$ this is because $\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_i\; =\; \backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_\{\backslash rho(i)\}$ and applying this condition (a) twice guarantees that both $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash mu\backslash left(F\_i\backslash right)\; =\; \backslash mu\backslash left(\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_i\backslash right)$ and $\backslash mu\backslash left(\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_\{\backslash rho(i)\}\backslash right)\; =\; \backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash mu\backslash left(F\_\{\backslash rho(i)\}\backslash right)$ 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 $F\_1,\; F\_2,\; \backslash ldots$ to the new order $F\_\{\backslash rho(1)\},\; F\_\{\backslash rho(2)\},\; \backslash ldots$ does not affect the sum of their measures. This is desirable since just as the union $F\; ~\backslash stackrel\{\backslash scriptscriptstyle\backslash text\{def\}\}\{=\}~\; \backslash textstyle\backslash bigcup\backslash limits\_\{i\; \backslash in\; \backslash N\}\; F\_i$ does not depend on the order of these sets, the same should be true of the sums $\backslash mu(F)\; =\; \backslash mu\backslash left(F\_1\backslash right)\; +\; \backslash mu\backslash left(F\_2\backslash right)\; +\; \backslash cdots$ and $\backslash mu(F)\; =\; \backslash mu\backslash left(F\_\{\backslash rho(1)\}\backslash right)\; +\; \backslash mu\backslash left(F\_\{\backslash rho(2)\}\backslash right)\; +\; \backslash cdots\backslash ,.$

  2. if $\backslash mu\backslash left(\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_i\backslash right)$ is not infinite then this series $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash mu\backslash left(F\_i\backslash right)$ must also converge absolutely, which by definition means that $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash left|\backslash mu\backslash left(F\_i\backslash right)\backslash right|$ must be finite. This is automatically true if $\backslash mu$ 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 $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash mu\backslash left(F\_i\backslash right)\; =\; \{\backslash displaystyle\backslash lim\_\{N\; \backslash to\; \backslash infty\}\}\; \backslash mu\backslash left(F\_1\backslash right)\; +\; \backslash mu\backslash left(F\_2\backslash right)\; +\; \backslash cdots\; +\; \backslash mu\backslash left(F\_N\backslash right)$ 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 $\backslash mu$ is valued in $[-\backslash infty,\; \backslash infty].$

  3. if $\backslash mu\backslash left(\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_i\backslash right)\; =\; \backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash mu\backslash left(F\_i\backslash right)$ is infinite then it is also required that the value of at least one of the series be finite (so that the sum of their values is well-defined). This is automatically true if $\backslash mu$ is non-negative.

• a Pre-measure if it is non-negative, countably additive (including finitely additive), and has a null empty set.
• a Measure (mathematics) 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.
• a Probability measure if it is a measure that has a mass of $1.$
• an Outer measure if it is non-negative, countably subadditive, has a null empty set, and has the power set $\backslash wp(\backslash Omega)$ as its domain.
  • Outer measures appear in the Carathéodory's extension theorem and they are often restricted to Carathéodory measurable subsets

• a Signed measure if it is countably additive, has a null empty set, and $\backslash mu$ does not take on both $-\; \backslash infty$ and $+\; \backslash infty$ as values.
• Complete measure if every subset of every null set is null; explicitly, this means: whenever $F\; \backslash in\; \backslash mathcal\{F\}\; \backslash text\{\; satisfies\; \}\; \backslash mu(F)\; =\; 0$ and $N\; \backslash subseteq\; F$ is any subset of $F$ then $N\; \backslash in\; \backslash mathcal\{F\}$ and $\backslash mu(N)\; =\; 0.$
  • Unlike many other properties, completeness places requirements on the set $\backslash operatorname\{domain\}\; \backslash mu\; =\; \backslash mathcal\{F\}$ (and not just on $\backslash mu$'s values).

• σ-finite measure if there exists a sequence $F\_1,\; F\_2,\; F\_3,\; \backslash ldots\backslash ,$ in $\backslash mathcal\{F\}$ such that $\backslash mu\backslash left(F\_i\backslash right)$ is finite for every index $i,$ and also $\backslash textstyle\backslash bigcup\backslash limits\_\{n=1\}^\backslash infty\; F\_n\; =\; \backslash textstyle\backslash bigcup\backslash limits\_\{F\; \backslash in\; \backslash mathcal\{F\}\}\; F.$
• Decomposable measure if there exists a subfamily $\backslash mathcal\{P\}\; \backslash subseteq\; \backslash mathcal\{F\}$ of pairwise disjoint sets such that $\backslash mu(P)$ is finite for every $P\; \backslash in\; \backslash mathcal\{P\}$ and also $\backslash textstyle\backslash bigcup\backslash limits\_\{P\; \backslash in\; \backslash mathcal\{P\}\}\; \backslash ,\; P\; =\; \backslash textstyle\backslash bigcup\backslash limits\_\{F\; \backslash in\; \backslash mathcal\{F\}\}\; F$ (where $\backslash mathcal\{F\}\; =\; \backslash operatorname\{domain\}\; \backslash mu$).
  • Every {{sigma}}-finite set function is decomposable although not conversely. For example, the counting measure on $\backslash R$ (whose domain is $\backslash wp(\backslash R)$) is decomposable but not {{sigma}}-finite.

• a Vector measure if it is a countably additive set function $\backslash mu\; :\; \backslash mathcal\{F\}\; \backslash to\; X$ valued in a topological vector space $X$ (such as a normed space) whose domain is a σ-algebra.
  • If $\backslash mu$ is valued in a normed space $(X,\; \backslash |\backslash cdot\backslash |)$ then it is countably additive if and only if for any pairwise disjoint sequence $F\_1,\; F\_2,\; \backslash ldots\backslash ,$ in $\backslash mathcal\{F\},$ $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash left\backslash |\backslash mu\backslash left(F\_1\backslash right)\; +\; \backslash cdots\; +\; \backslash mu\backslash left(F\_n\backslash right)\; -\; \backslash mu\backslash left(\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_i\backslash right)\backslash right\backslash |\; =\; 0.$ If $\backslash mu$ is finitely additive and valued in a Banach space then it is countably additive if and only if for any pairwise disjoint sequence $F\_1,\; F\_2,\; \backslash ldots\backslash ,$ in $\backslash mathcal\{F\},$ $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash left\backslash |\backslash mu\backslash left(F\_n\; \backslash cup\; F\_\{n+1\}\; \backslash cup\; F\_\{n+2\}\; \backslash cup\; \backslash cdots\backslash right)\backslash right\backslash |\; =\; 0.$

• a Complex measure if it is a countably additive complex-valued set function $\backslash mu\; :\; \backslash mathcal\{F\}\; \backslash to\; \backslash Complex$ whose domain is a σ-algebra.
  • By definition, a complex measure never takes $\backslash pm\; \backslash infty$ as a value and so has a null empty set.

• a Random measure if it is a measure-valued random element.

Arbitrary sums

As described in this article's section on generalized series, for any family $\backslash left(r\_i\backslash right)\_\{i\; \backslash in\; I\}$ of real numbers indexed by an arbitrary indexing set $I,$ it is possible to define their sum $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i$ as the limit of the net of finite partial sums $F\; \backslash in\; \backslash operatorname\{FiniteSubsets\}(I)\; \backslash mapsto\; \backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; F\}\; r\_i$ where the domain $\backslash operatorname\{FiniteSubsets\}(I)$ is directed by $\backslash ,\backslash subseteq.\backslash ,$

Whenever this net converges then its limit is denoted by the symbols $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i$ while if this net instead diverges to $\backslash pm\; \backslash infty$ then this may be indicated by writing $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i\; =\; \backslash pm\; \backslash infty.$

Any sum over the empty set is defined to be zero; that is, if $I\; =\; \backslash varnothing$ then $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; \backslash varnothing\}\; r\_i\; =\; 0$ by definition.

For example, if $z\_i\; =\; 0$ for every $i\; \backslash in\; I$ then $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; z\_i\; =\; 0.$

And it can be shown that $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i\; =\; \backslash textstyle\backslash sum\backslash limits\_\{\backslash stackrel\{i\; \backslash in\; I,\}\{r\_i\; =\; 0\}\}\; r\_i\; +\; \backslash textstyle\backslash sum\backslash limits\_\{\backslash stackrel\{i\; \backslash in\; I,\}\{r\_i\; \backslash neq\; 0\}\}\; r\_i\; =\; 0\; +\; \backslash textstyle\backslash sum\backslash limits\_\{\backslash stackrel\{i\; \backslash in\; I,\}\{r\_i\; \backslash neq\; 0\}\}\; r\_i\; =\; \backslash textstyle\backslash sum\backslash limits\_\{\backslash stackrel\{i\; \backslash in\; I,\}\{r\_i\; \backslash neq\; 0\}\}\; r\_i.$

If $I\; =\; \backslash N$ then the generalized series $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i$ converges in $\backslash R$ if and only if $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; r\_i$ converges unconditionally (or equivalently, converges absolutely) in the usual sense.

If a generalized series $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i$ converges in $\backslash R$ then both $\backslash textstyle\backslash sum\backslash limits\_\{\backslash stackrel\{i\; \backslash in\; I\}\{r\_i\; >\; 0\}\}\; r\_i$ and also converge to elements of $\backslash R$ and the set $\backslash left\backslash \{i\; \backslash in\; I\; :\; r\_i\; \backslash neq\; 0\backslash right\backslash \}$ is necessarily countable (that is, either finite or countably infinite); this remains true if $\backslash R$ is replaced with any normed space.

It follows that in order for a generalized series $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i$ to converge in $\backslash R$ or $\backslash Complex,$ it is necessary that all but at most countably many $r\_i$ will be equal to $0,$ which means that $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i\; ~=~\; \backslash textstyle\backslash sum\backslash limits\_\{\backslash stackrel\{i\; \backslash in\; I\}\{r\_i\; \backslash neq\; 0\}\}\; r\_i$ is a sum of at most countably many non-zero terms.

Said differently, if $\backslash left\backslash \{i\; \backslash in\; I\; :\; r\_i\; \backslash neq\; 0\backslash right\backslash \}$ is uncountable then the generalized series $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i$ 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 $F\_1,\; F\_2,\; \backslash ldots\backslash ,$ in $\backslash mathcal\{F\}$ (and the usual countable series $\backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash mu\backslash left(F\_i\backslash right)$) to arbitrarily many sets $\backslash left(F\_i\backslash right)\_\{i\; \backslash in\; I\}$ (and the generalized series $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; \backslash mu\backslash left(F\_i\backslash right)$).

A set function $\backslash mu$ is said to be/satisfies{{sfn|Durrett|2019|pp=1-37, 455-470}}

• {{em|{{visible anchor|monotone}}}} if $\backslash mu(E)\; \backslash leq\; \backslash mu(F)$ whenever $E,\; F\; \backslash in\; \backslash mathcal\{F\}$ satisfy $E\; \backslash subseteq\; F.$
• Modular set function if it satisfies the following condition, known as {{em|modularity}}: $\backslash mu(E\; \backslash cup\; F)\; +\; \backslash mu(E\; \backslash cap\; F)\; =\; \backslash mu(E)\; +\; \backslash mu(F)$ for all $E,\; F\; \backslash in\; \backslash mathcal\{F\}$ such that $E\; \backslash cup\; F,\; E\; \backslash cap\; F\; \backslash in\; \backslash mathcal\{F\}.$
  • 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 Valuation (geometry). This geometric definition of "valuation" should not be confused with the stronger non-equivalent measure theoretic definition of "valuation" that is given below.

• Submodular set function if $\backslash mu(E\; \backslash cup\; F)\; +\; \backslash mu(E\; \backslash cap\; F)\; \backslash leq\; \backslash mu(E)\; +\; \backslash mu(F)$ for all $E,\; F\; \backslash in\; \backslash mathcal\{F\}$ such that $E\; \backslash cup\; F,\; E\; \backslash cap\; F\; \backslash in\; \backslash mathcal\{F\}.$
• {{em|{{visible anchor|finitely subadditive}}}} if $|\backslash mu(F)|\; \backslash leq\; \backslash textstyle\backslash sum\backslash limits\_\{i=1\}^n\; \backslash left|\backslash mu\backslash left(F\_i\backslash right)\backslash right|$ for all finite sequences $F,\; F\_1,\; \backslash ldots,\; F\_n\; \backslash in\; \backslash mathcal\{F\}$ that satisfy $F\; \backslash ;\backslash subseteq\backslash ;\; \backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^n\; F\_i.$
• {{em|{{visible anchor|countably subadditive}}}} or {{em|{{visible anchor|σ-subadditive}}}} if $|\backslash mu(F)|\; \backslash leq\; \backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash left|\backslash mu\backslash left(F\_i\backslash right)\backslash right|$ for all sequences $F,\; F\_1,\; F\_2,\; F\_3,\; \backslash ldots\backslash ,$ in $\backslash mathcal\{F\}$ that satisfy $F\; \backslash ;\backslash subseteq\backslash ;\; \backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_i.$
  • If $\backslash mathcal\{F\}$ is closed under finite unions then this condition holds if and only if $|\backslash mu(F\; \backslash cup\; G)|\; \backslash leq|\; \backslash mu(F)|\; +\; |\backslash mu(G)|$ for all $F,\; G\; \backslash in\; \backslash mathcal\{F\}.$ If $\backslash mu$ is non-negative then the absolute values may be removed.
  • If $\backslash mu$ is a measure then this condition holds if and only if $\backslash mu\backslash left(\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_i\backslash right)\; \backslash leq\; \backslash textstyle\backslash sum\backslash limits\_\{i=1\}^\backslash infty\; \backslash mu\backslash left(F\_i\backslash right)$ for all $F\_1,\; F\_2,\; F\_3,\; \backslash ldots\backslash ,$ in $\backslash mathcal\{F\}.${{sfn|Royden|Fitzpatrick|2010|p=30}} If $\backslash mu$ is a probability measure then this inequality is Boole's inequality.
  • If $\backslash mu$ is countably subadditive and $\backslash varnothing\; \backslash in\; \backslash mathcal\{F\}$ with $\backslash mu(\backslash varnothing)\; =\; 0$ then $\backslash mu$ is finitely subadditive.

• Superadditivity if $\backslash mu(E)\; +\; \backslash mu(F)\; \backslash leq\; \backslash mu(E\; \backslash cup\; F)$ whenever $E,\; F\; \backslash in\; \backslash mathcal\{F\}$ are disjoint with $E\; \backslash cup\; F\; \backslash in\; \backslash mathcal\{F\}.$
• {{em|{{visible anchor|continuous from above}}}} if $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash mu\backslash left(F\_i\backslash right)\; =\; \backslash mu\backslash left(\backslash textstyle\backslash bigcap\backslash limits\_\{i=1\}^\backslash infty\; F\_i\backslash right)$ for all {{em|non-increasing sequences}} of sets $F\_1\; \backslash supseteq\; F\_2\; \backslash supseteq\; F\_3\; \backslash cdots\backslash ,$ in $\backslash mathcal\{F\}$ such that $\backslash textstyle\backslash bigcap\backslash limits\_\{i=1\}^\backslash infty\; F\_i\; \backslash in\; \backslash mathcal\{F\}$ with $\backslash mu\backslash left(\backslash textstyle\backslash bigcap\backslash limits\_\{i=1\}^\backslash infty\; F\_i\backslash right)$ and all $\backslash mu\backslash left(F\_i\backslash right)$ finite.
  • Lebesgue measure $\backslash lambda$ is continuous from above but it would not be if the assumption that all $\backslash mu\backslash left(F\_i\backslash right)$ are eventually finite was omitted from the definition, as this example shows: For every integer $i,$ let $F\_i$ be the open interval $(i,\; \backslash infty)$ so that $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash lambda\backslash left(F\_i\backslash right)\; =\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash infty\; =\; \backslash infty\; \backslash neq\; 0\; =\; \backslash lambda(\backslash varnothing)\; =\; \backslash lambda\backslash left(\backslash textstyle\backslash bigcap\backslash limits\_\{i=1\}^\backslash infty\; F\_i\backslash right)$ where $\backslash textstyle\backslash bigcap\backslash limits\_\{i=1\}^\backslash infty\; F\_i\; =\; \backslash varnothing.$

• {{em|{{visible anchor|continuous from below}}}} if $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash mu\backslash left(F\_i\backslash right)\; =\; \backslash mu\backslash left(\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_i\backslash right)$ for all {{em|non-decreasing sequences}} of sets $F\_1\; \backslash subseteq\; F\_2\; \backslash subseteq\; F\_3\; \backslash cdots\backslash ,$ in $\backslash mathcal\{F\}$ such that $\backslash textstyle\backslash bigcup\backslash limits\_\{i=1\}^\backslash infty\; F\_i\; \backslash in\; \backslash mathcal\{F\}.$
• {{em|{{visible anchor|infinity is approached from below}}}} if whenever $F\; \backslash in\; \backslash mathcal\{F\}$ satisfies $\backslash mu(F)\; =\; \backslash infty$ then for every real $r\; >\; 0,$ there exists some $F\_r\; \backslash in\; \backslash mathcal\{F\}$ such that $F\_r\; \backslash subseteq\; F$ and
• an #outer measure if $\backslash mu$ is non-negative, countably subadditive, has a null empty set, and has the power set $\backslash wp(\backslash Omega)$ as its domain.
• an Inner measure if $\backslash mu$ is non-negative, superadditive, continuous from above, has a null empty set, has the power set $\backslash wp(\backslash Omega)$ as its domain, and $+\backslash infty$ is approached from below.
• Atomic measure if every measurable set of positive measure contains an atom.

If a binary operation $\backslash ,+\backslash ,$ is defined, then a set function $\backslash mu$ is said to be

• {{em|Translation invariant}} if $\backslash mu(\backslash omega\; +\; F)\; =\; \backslash mu(F)$ for all $\backslash omega\; \backslash in\; \backslash Omega$ and $F\; \backslash in\; \backslash mathcal\{F\}$ such that $\backslash omega\; +\; F\; \backslash in\; \backslash mathcal\{F\}.$

If $\backslash tau$ is a topology on $\backslash Omega$ then a set function $\backslash mu$ is said to be:

• a 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 $\backslash tau$).
• a Baire measure if it is a measure defined on the σ-algebra of all Baire sets.
• Locally finite measure if for every point $\backslash omega\; \backslash in\; \backslash Omega$ there exists some neighborhood $U\; \backslash in\; \backslash mathcal\{F\}\; \backslash cap\; \backslash tau$ of this point such that $\backslash mu(U)$ is finite.
  • If $\backslash mu$ is a finitely additive, monotone, and locally finite then $\backslash mu(K)$ is necessarily finite for every compact measurable subset $K.$

• τ-additivity if $\backslash mu\backslash left(\{\backslash textstyle\backslash bigcup\}\; \backslash ,\; \backslash mathcal\{D\}\backslash right)\; =\; \backslash sup\_\{D\; \backslash in\; \backslash mathcal\{D\}\}\; \backslash mu(D)$ whenever $\backslash mathcal\{D\}\; \backslash subseteq\; \backslash tau\; \backslash cap\; \backslash mathcal\{F\}$ is directed with respect to $\backslash ,\backslash subseteq\backslash ,$ and satisfies $\{\backslash textstyle\backslash bigcup\}\; \backslash ,\; \backslash mathcal\{D\}\; ~\backslash stackrel\{\backslash scriptscriptstyle\backslash text\{def\}\}\{=\}~\; \backslash textstyle\backslash bigcup\backslash limits\_\{D\; \backslash in\; \backslash mathcal\{D\}\}\; D\; \backslash in\; \backslash mathcal\{F\}.$
  • $\backslash mathcal\{D\}$ is directed with respect to $\backslash ,\backslash subseteq\backslash ,$ if and only if it is not empty and for all $A,\; B\; \backslash in\; \backslash mathcal\{D\}$ there exists some $C\; \backslash in\; \backslash mathcal\{D\}$ such that $A\; \backslash subseteq\; C$ and $B\; \backslash subseteq\; C.$

• Inner regular measure or {{em|{{visible anchor|tight}}}} if for every $F\; \backslash in\; \backslash mathcal\{F\},$ $\backslash mu(F)\; =\; \backslash sup\; \backslash \{\backslash mu(K)\; :\; F\; \backslash supseteq\; K\; \backslash text\{\; with\; \}\; K\; \backslash in\; \backslash mathcal\{F\}\; \backslash text\{\; a\; compact\; subset\; of\; \}\; (\backslash Omega,\; \backslash tau)\backslash \}.$
• Outer regular measure if for every $F\; \backslash in\; \backslash mathcal\{F\},$ $\backslash mu(F)\; =\; \backslash inf\; \backslash \{\backslash mu(U)\; :\; F\; \backslash subseteq\; U\; \backslash text\{\; and\; \}\; U\; \backslash in\; \backslash mathcal\{F\}\; \backslash cap\; \backslash tau\backslash \}.$
• Regular measure if it is both inner regular and outer regular.
• a Borel regular measure if it is a Borel measure that is also Regular measure.
• a Radon measure if it is a regular and locally finite measure.
• Strictly positive measure if every non-empty open subset has (strictly) positive measure.
• a Valuation (measure theory) if it is non-negative, monotone, modular, has a null empty set, and has domain $\backslash tau.$

{{See also|Radon–Nikodym theorem|Lebesgue's decomposition theorem}}

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

• $\backslash mu$ is said to be Absolute continuity (measure theory) or Domination (measure theory), written $\backslash mu\; \backslash ll\; \backslash nu,$ if for every set $F$ that belongs to the domain of both $\backslash mu$ and $\backslash nu,$ if $\backslash nu(F)\; =\; 0$ then $\backslash mu(F)\; =\; 0.$
  • If $\backslash mu$ and $\backslash nu$ are $\backslash sigma$-finite measures on the same measurable space and if $\backslash mu\; \backslash ll\; \backslash nu,$ then the Radon–Nikodym derivative $\backslash frac\{d\; \backslash mu\}\{d\; \backslash nu\}$ exists and for every measurable $F,$ $$\backslash mu(F)\; =\; \backslash int\_F\; \backslash frac\{d\; \backslash mu\}\{d\; \backslash nu\}\; d\; \backslash nu.$$
  • $\backslash mu$ and $\backslash nu$ are called Equivalence (measure theory) if each one is absolutely continuous with respect to the other. $\backslash mu$ is called a Equivalence (measure theory)#Supporting measure of a measure $\backslash nu$ if $\backslash mu$ is $\backslash sigma$-finite and they are equivalent.{{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}}

• $\backslash mu$ and $\backslash nu$ are Singular measure, written $\backslash mu\; \backslash perp\; \backslash nu,$ if there exist disjoint sets $M$ and $N$ in the domains of $\backslash mu$ and $\backslash nu$ such that $M\; \backslash cup\; N\; =\; \backslash Omega,$ $\backslash mu(F)\; =\; 0$ for all $F\; \backslash subseteq\; M$ in the domain of $\backslash mu,$ and $\backslash nu(F)\; =\; 0$ for all $F\; \backslash subseteq\; N$ in the domain of $\backslash nu.$

Examples of set functions include:

• The function $$d(A)\; =\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; \backslash frac$$

  A \cap \{1, \ldots, n\}

  {n}, assigning densities to sufficiently well-behaved subsets $A\; \backslash subseteq\; \backslash \{1,\; 2,\; 3,\; \backslash ldots\backslash \},$ 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 $1,$ 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 {{em|random set}} is a set-valued random variable. See the article random compact set.

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

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

Its definition begins with the set $\backslash operatorname\{Intervals\}(\backslash Reals)$ of all intervals of real numbers, which is a semialgebra on $\backslash Reals.$

The function that assigns to every interval $I$ its $\backslash operatorname\{length\}(I)$ is a finitely additive set function (explicitly, if $I$ has endpoints $a\; \backslash leq\; b$ then $\backslash operatorname\{length\}(I)\; =\; b\; -\; a$).

This set function can be extended to the Lebesgue outer measure on $\backslash Reals,$ which is the translation-invariant set function $\backslash lambda^\{\backslash !*\backslash !\}\; :\; \backslash wp(\backslash Reals)\; \backslash to\; [0,\; \backslash infty]$ that sends a subset $E\; \backslash subseteq\; \backslash Reals$ to the infimum

$$\backslash lambda^\{\backslash !*\backslash !\}(E)\; =\; \backslash inf\; \backslash left\backslash \{\backslash sum\_\{k=1\}^\backslash infty\; \backslash operatorname\{length\}(I\_k)\; :\; \{(I\_k)\_\{k\; \backslash in\; \backslash N\}\}\; \backslash text\{\; is\; a\; sequence\; of\; open\; intervals\; with\; \}\; E\; \backslash subseteq\; \backslash bigcup\_\{k=1\}^\backslash infty\; I\_k\backslash right\backslash \}.$$

Lebesgue outer measure is not countably additive (and so is not a measure) although its restriction to the {{sigma}}-algebra of all subsets $M\; \backslash subseteq\; \backslash Reals$ that satisfy the Carathéodory criterion:

$$\backslash lambda^\{\backslash !*\backslash !\}(M)\; =\; \backslash lambda^\{\backslash !*\backslash !\}(M\; \backslash cap\; E)\; +\; \backslash lambda^\{\backslash !*\backslash !\}(M\; \backslash cap\; E^c)\; \backslash quad\; \backslash text\{\; for\; every\; \}\; S\; \backslash subseteq\; \backslash Reals$$

is a measure that called Lebesgue measure.

Vitali sets are examples of non-measurable sets of real numbers.

{{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.

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

{{Math theorem

| name = Theorem{{sfn|Rudin|1991|pp=141-142}}

| math_statement =

If $(G,\; +)$ is any abelian group then there exists a finitely additive and translation-invariantThe function $\backslash mu$ being translation-invariant means that $\backslash mu(S)\; =\; \backslash mu(g\; +\; S)$ for every $g\; \backslash in\; G$ and every subset $S\; \backslash subseteq\; G.$ set function $\backslash mu\; :\; \backslash wp(G)\; \backslash to\; [0,\; 1]$ of mass $\backslash mu(G)\; =\; 1.$

}}

{{See also|Carathéodory's extension theorem}}

Suppose that $\backslash mu$ is a set function on a semialgebra $\backslash mathcal\{F\}$ over $\backslash Omega$ and let

$$\backslash operatorname\{algebra\}(\backslash mathcal\{F\})\; :=\; \backslash left\backslash \{\; F\_1\; \backslash sqcup\; \backslash cdots\; \backslash sqcup\; F\_n\; :\; n\; \backslash in\; \backslash N\; \backslash text\{\; and\; \}\; F\_1,\; \backslash ldots,\; F\_n\; \backslash in\; \backslash mathcal\{F\}\; \backslash text\{\; are\; pairwise\; disjoint\; \}\; \backslash right\backslash \},$$

which is the algebra on $\backslash Omega$ generated by $\backslash mathcal\{F\}.$

The archetypal example of a semialgebra that is not also an algebra is the family

on $\backslash Omega\; :=\; \backslash R^d$ where for all {{sfn|Durrett|2019|pp=1-9}} Importantly, the two non-strict inequalities $\backslash ,\backslash leq\backslash ,$ in cannot be replaced with strict inequalities since semialgebras must contain the whole underlying set $\backslash R^d;$ that is, $\backslash R^d\; \backslash in\; \backslash mathcal\{S\}\_d$ is a requirement of semialgebras (as is $\backslash varnothing\; \backslash in\; \backslash mathcal\{S\}\_d$).

If $\backslash mu$ is finitely additive then it has a unique extension to a set function $\backslash overline\{\backslash mu\}$ on $\backslash operatorname\{algebra\}(\backslash mathcal\{F\})$ defined by sending $F\_1\; \backslash sqcup\; \backslash cdots\; \backslash sqcup\; F\_n\; \backslash in\; \backslash operatorname\{algebra\}(\backslash mathcal\{F\})$ (where $\backslash ,\backslash sqcup\backslash ,$ indicates that these $F\_i\; \backslash in\; \backslash mathcal\{F\}$ are pairwise disjoint) to:{{sfn|Durrett|2019|pp=1-9}}

$$\backslash overline\{\backslash mu\}\backslash left(F\_1\; \backslash sqcup\; \backslash cdots\; \backslash sqcup\; F\_n\backslash right)\; :=\; \backslash mu\backslash left(F\_1\backslash right)\; +\; \backslash cdots\; +\; \backslash mu\backslash left(F\_n\backslash right).$$

This extension $\backslash overline\{\backslash mu\}$ will also be finitely additive: for any pairwise disjoint $A\_1,\; \backslash ldots,\; A\_n\; \backslash in\; \backslash operatorname\{algebra\}(\backslash mathcal\{F\}),$ {{sfn|Durrett|2019|pp=1-9}}

$$\backslash overline\{\backslash mu\}\backslash left(A\_1\; \backslash cup\; \backslash cdots\; \backslash cup\; A\_n\backslash right)\; =\; \backslash overline\{\backslash mu\}\backslash left(A\_1\backslash right)\; +\; \backslash cdots\; +\; \backslash overline\{\backslash mu\}\backslash left(A\_n\backslash right).$$

If in addition $\backslash mu$ is extended real-valued and monotone (which, in particular, will be the case if $\backslash mu$ is non-negative) then $\backslash overline\{\backslash mu\}$ will be monotone and finitely subadditive: for any $A,\; A\_1,\; \backslash ldots,\; A\_n\; \backslash in\; \backslash operatorname\{algebra\}(\backslash mathcal\{F\})$ such that $A\; \backslash subseteq\; A\_1\; \backslash cup\; \backslash cdots\; \backslash cup\; A\_n,${{sfn|Durrett|2019|pp=1-9}}

$$\backslash overline\{\backslash mu\}\backslash left(A\backslash right)\; \backslash leq\; \backslash overline\{\backslash mu\}\backslash left(A\_1\backslash right)\; +\; \backslash cdots\; +\; \backslash overline\{\backslash mu\}\backslash left(A\_n\backslash right).$$

{{See also|Pre-measure|Hahn–Kolmogorov theorem}}

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

To define this extension, first extend $\backslash mu$ to an outer measure $\backslash mu^*$ on $2^\backslash Omega\; =\; \backslash wp(\backslash Omega)$ by

$$\backslash mu^*(T)\; =\; \backslash inf\; \backslash left\backslash \{\backslash sum\_n\; \backslash mu\backslash left(S\_n\backslash right)\; :\; T\; \backslash subseteq\; \backslash cup\_n\; S\_n\; \backslash text\{\; with\; \}\; S\_1,\; S\_2,\; \backslash ldots\; \backslash in\; \backslash mathcal\{F\}\backslash right\backslash \}$$

and then restrict it to the set $\backslash mathcal\{F\}\_M$ of $\backslash mu^*$-measurable sets (that is, Carathéodory-measurable sets), which is the set of all $M\; \backslash subseteq\; \backslash Omega$ such that

$$\backslash mu^*(S)\; =\; \backslash mu^*(S\; \backslash cap\; M)\; +\; \backslash mu^*(S\; \backslash cap\; M^\backslash mathrm\{c\})\; \backslash quad\; \backslash text\{\; for\; every\; subset\; \}\; S\; \backslash subseteq\; \backslash Omega.$$ It is a $\backslash sigma$-algebra and $\backslash mu^*$ is sigma-additive on it, by Caratheodory lemma.

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

If $\backslash mu^*\; :\; \backslash wp(\backslash Omega)\; \backslash to\; [0,\; \backslash infty]$ is an outer measure on a set $\backslash Omega,$ where (by definition) the domain is necessarily the power set $\backslash wp(\backslash Omega)$ of $\backslash Omega,$ then a subset $M\; \backslash subseteq\; \backslash Omega$ is called {{em|$\backslash mu^*$–measurable}} or {{em|Carathéodory-measurable}} if it satisfies the following {{em|Carathéodory's criterion}}:

$$\backslash mu^*(S)\; =\; \backslash mu^*(S\; \backslash cap\; M)\; +\; \backslash mu^*(S\; \backslash cap\; M^\backslash mathrm\{c\})\; \backslash quad\; \backslash text\{\; for\; every\; subset\; \}\; S\; \backslash subseteq\; \backslash Omega,$$

where $M^\backslash mathrm\{c\}\; :=\; \backslash Omega\; \backslash setminus\; M$ is the complement of $M.$

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

• {{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}}

{{reflist}}

{{reflist|group=note}}

Proofs

{{reflist|group=proof|refs=

Suppose the net $\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; I\}\; r\_i\; ~\backslash stackrel\{\backslash scriptscriptstyle\backslash text\{def\}\}\{=\}~\; \{\backslash textstyle\backslash lim\backslash limits\_\{A\; \backslash in\; \backslash operatorname\{Finite\}(I)\}\}\; \backslash \; \backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; A\}\; r\_i\; =\; \backslash lim\; \backslash left\backslash \{\backslash textstyle\backslash sum\backslash limits\_\{i\backslash in\; A\}\; r\_i\; \backslash ,:\; A\; \backslash subseteq\; I,\; A\; \backslash text\{\; finite\; \}\backslash right\backslash \}$ converges to some point in a metrizable topological vector space $X$ (such as $\backslash Reals,$ $\backslash Complex,$ or a normed space), where recall that this net's domain is the directed set $(\backslash operatorname\{Finite\}(I),\; \backslash subseteq).$

Like every convergent net, this convergent net of partial sums $A\; \backslash mapsto\; \backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; A\}\; r\_i$ is a {{em|Cauchy net}}, which for this particular net means (by definition) that for every neighborhood $W$ of the origin in $X,$ there exists a finite subset $A\_0$ of $I$ such that

$\backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; B\}\; r\_i\; -\; \backslash textstyle\backslash sum\backslash limits\_\{i\; \backslash in\; C\}\; r\_i\; \backslash in\; W$ for all finite supersets $B,\; C\; \backslash supseteq\; A\_0;$

this implies that $r\_i\; \backslash in\; W$ for every $i\; \backslash in\; I\; \backslash setminus\; A\_0$ (by taking $B\; :=\; A\_0\; \backslash cup\; \backslash \{i\backslash \}$ and $C\; :=\; A\_0$).

Since $X$ is metrizable, it has a countable neighborhood basis $U\_1,\; U\_2,\; \backslash ldots$ at the origin, whose intersection is necessarily $U\_1\; \backslash cap\; U\_2\; \backslash cap\; \backslash cdots\; =\; \backslash \{0\backslash \}$ (since $X$ is a Hausdorff TVS).

For every positive integer $n\; \backslash in\; \backslash N,$ pick a finite subset $A\_n\; \backslash subseteq\; I$ such that $r\_i\; \backslash in\; U\_n$ for every $i\; \backslash in\; I\; \backslash setminus\; A\_n.$

If $i$ belongs to $(I\; \backslash setminus\; A\_1)\; \backslash cap\; (I\; \backslash setminus\; A\_2)\; \backslash cap\; \backslash cdots\; =\; I\; \backslash setminus\; \backslash left(A\_1\; \backslash cup\; A\_2\; \backslash cup\; \backslash cdots\backslash right)$ then $r\_i$ belongs to $U\_1\; \backslash cap\; U\_2\; \backslash cap\; \backslash cdots\; =\; \backslash \{0\backslash \}.$

Thus $r\_i\; =\; 0$ for every index $i\; \backslash in\; I$ that does not belong to the countable set $A\_1\; \backslash cup\; A\_2\; \backslash cup\; \backslash cdots.$ $\backslash blacksquare$

}}

{{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}}

• {{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]

{{Measure theory}}

{{Analysis in topological vector spaces}}

Category:Basic concepts in set theory

Category:Functions and mappings

Category:Measure theory

Category:Measures (measure theory)