# Set function

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 ${\displaystyle \mathbb {R} \cup \{\pm \infty \},}$ which consists of the real numbers ${\displaystyle \mathbb {R} }$ and ${\displaystyle \pm \infty .}$

A set function generally aims to 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 ${\displaystyle {\mathcal {F}}}$ is a family of sets over ${\displaystyle \Omega }$ then a set function on ${\displaystyle {\mathcal {F}}}$ is a function ${\displaystyle \mu }$ with domain ${\displaystyle {\mathcal {F}}}$ and codomain ${\displaystyle [-\infty ,\infty ]}$ or, sometimes, the codomain is instead some vector space, as with vector measures, complex measures, and projection-valued measures. The total variation of a set ${\displaystyle S}$ is

${\displaystyle |\mu |(S):=\sup\{|\mu (F)|:F\in {\mathcal {F}}{\text{ and }}F\subseteq S\}}$
where ${\displaystyle |\,\cdot \,|}$ denotes the absolute value (or more generally, it denotes the norm or seminorm if ${\displaystyle \mu }$ is vector-valued in a (semi)normed space). Assuming that ${\displaystyle \cup {\mathcal {F}}:=\bigcup _{F\in {\mathcal {F}}}F\in {\mathcal {F}},}$ then ${\displaystyle |\mu |\left(\cup {\mathcal {F}}\right)}$ is called the total variation of ${\displaystyle \mu }$ and ${\displaystyle \mu \left(\cup {\mathcal {F}}\right)}$ is called the mass of ${\displaystyle \mu .}$ A set function is called finite if for every ${\displaystyle F\in {\mathcal {F}},}$ ${\displaystyle \mu (F)}$ is finite (that is, not equal to ${\displaystyle \pm \infty }$); every finite set function must have a finite mass.

In general, it is typically assumed that ${\displaystyle \mu (E)+\mu (F)}$ is always well-defined for all ${\displaystyle E,F\in {\mathcal {F}},}$ or equivalently, that ${\displaystyle \mu }$ does not take on both ${\displaystyle -\infty }$ and ${\displaystyle +\infty }$ 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 ${\displaystyle \mu }$ is finitely additive:

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

A set ${\displaystyle F\in {\mathcal {F}}}$ is called a null set (with respect to ${\displaystyle \mu }$) or simply null if ${\displaystyle \mu (F)=0.}$ Whenever ${\displaystyle \mu }$ is not identically equal to either ${\displaystyle -\infty }$ or ${\displaystyle +\infty }$ then it is typically also assumed that:

• null empty set: ${\displaystyle \mu (\varnothing )=0}$ if ${\displaystyle \varnothing \in {\mathcal {F}}.}$

### Common properties of set functions

A set function ${\displaystyle \mu }$ on ${\displaystyle {\mathcal {F}}}$ is said to be[1]

• non-negative if it is valued in ${\displaystyle [0,\infty ].}$
• finitely additive if ${\displaystyle \sum _{i=1}^{n}\mu \left(F_{i}\right)=\mu \left(\bigcup _{i=1}^{n}F_{i}\right)}$ for all pairwise disjoint finite sequences ${\displaystyle F_{1},\ldots ,F_{n}\in {\mathcal {F}}}$ such that ${\displaystyle \bigcup _{i=1}^{n}F_{i}\in {\mathcal {F}}.}$
• If ${\displaystyle {\mathcal {F}}}$ is closed under binary unions then ${\displaystyle \mu }$ is finitely additive if and only if ${\displaystyle \mu (E\cup F)=\mu (E)+\mu (F)}$ for all disjoint pairs ${\displaystyle E,F\in {\mathcal {F}}.}$
• If ${\displaystyle \mu }$ is finitely additive and if ${\displaystyle \varnothing \in {\mathcal {F}}}$ then taking ${\displaystyle E:=F:=\varnothing }$ shows that ${\displaystyle \mu (\varnothing )=\mu (\varnothing )+\mu (\varnothing )}$ which is only possible if ${\displaystyle \mu (\varnothing )=0}$ or ${\displaystyle \mu (\varnothing )=\pm \infty ,}$ where in the latter case, ${\displaystyle \mu (E)=\mu (E\cup \varnothing )=\mu (E)+\mu (\varnothing )=\mu (E)+(\pm \infty )=\pm \infty }$ for every ${\displaystyle E\in {\mathcal {F}}}$ (so only the case ${\displaystyle \mu (\varnothing )=0}$ is useful).
• countably additive or σ-additive[2] if it is finitely additive and also ${\displaystyle \sum _{i=1}^{\infty }\mu \left(F_{i}\right)=\mu \left(\bigcup _{i=1}^{\infty }F_{i}\right)}$ for all pairwise disjoint sequences ${\displaystyle F_{1},F_{2},\ldots \,}$ in ${\displaystyle {\mathcal {F}}}$ such that ${\displaystyle \bigcup _{i=1}^{\infty }F_{i}\in {\mathcal {F}}}$ and where it is also required that:
1. if ${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }F_{i}\right)}$ is not infinite then this series ${\displaystyle \sum _{i=1}^{\infty }\mu \left(F_{i}\right)}$ must also converge absolutely, which by definition means that ${\displaystyle \sum _{i=1}^{\infty }\left|\mu \left(F_{i}\right)\right|}$ must be finite. This is automatically true if ${\displaystyle \mu }$ is non-negative.
• As with any convergent series of real numbers, by the Riemann series theorem, the series ${\displaystyle \sum _{i=1}^{\infty }\mu \left(F_{i}\right):=\lim _{N\to \infty }\mu \left(F_{1}\right)+\mu \left(F_{2}\right)+\cdots \mu \left(F_{N}\right)}$ converges absolutely if and only if its sum does not depend on the order of its terms (a property known as unconditional convergence).
2. if ${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }F_{i}\right)=\sum _{i=1}^{\infty }\mu \left(F_{i}\right)}$ is infinite then it is also required that the value of at least one of the series ${\displaystyle \sum _{\stackrel {i\in \mathbb {N} }{\mu \left(F_{i}\right)>0}}\mu \left(F_{i}\right)\;{\text{ and }}\;\sum _{\stackrel {i\in \mathbb {N} }{\mu \left(F_{i}\right)<0}}\mu \left(F_{i}\right)\;}$ be finite (so that the sum of their values is well-defined). This is automatically true if ${\displaystyle \mu }$ is non-negative.
• Assuming that 𝜎-additivity holds, then the series ${\displaystyle \sum _{i=1}^{\infty }\mu \left(F_{i}\right)}$ must be unconditionally convergent, which means that if ${\displaystyle \rho :\mathbb {N} \to \mathbb {N} }$ is any permutation/bijection then ${\displaystyle \sum _{i=1}^{\infty }\mu \left(F_{i}\right)=\sum _{i=1}^{\infty }\mu \left(F_{\rho (i)}\right)}$ (this is true because ${\displaystyle \sum _{i=1}^{\infty }\mu \left(F_{i}\right)=\mu \left(\bigcup _{i=1}^{\infty }F_{i}\right)}$ and also ${\displaystyle \mu \left(\bigcup _{i=1}^{\infty }F_{\rho (i)}\right)=\sum _{i=1}^{\infty }\mu \left(F_{\rho (i)}\right)}$ where ${\displaystyle \bigcup _{i=1}^{\infty }F_{i}=\bigcup _{i=1}^{\infty }F_{\rho (i)}}$). This guarantees that relabeling/rearranging the sets ${\displaystyle F_{1},F_{2},\ldots }$ to the new order ${\displaystyle F_{\rho (1)},F_{\rho (2)},\ldots }$ does not affect the sum of their measures. This is a reasonable since just as the union of ${\displaystyle F:=\bigcup _{i\in \mathbb {N} }F_{i}}$ does not depend on the order of these sets, the same should be true of the sums ${\displaystyle \mu (F)=\mu \left(F_{1}\right)+\mu \left(F_{2}\right)+\cdots }$ and ${\displaystyle \mu (F)=\mu \left(F_{\rho (1)}\right)+\mu \left(F_{\rho (2)}\right)+\cdots \,.}$
• a measure if ${\displaystyle \mu }$ is non-negative, countably additive, and has a null empty set.
• a signed measure if ${\displaystyle \mu }$ is countably additive, has a null empty set, and ${\displaystyle \mu }$ does not take on both ${\displaystyle -\infty }$ and ${\displaystyle +\infty }$ as values.
• a probability measure if it is a measure that has a mass of ${\displaystyle 1.}$
• complete if whenever ${\displaystyle F\in {\mathcal {F}}{\text{ satisfies }}\mu (F)=0{\text{ and }}N\subseteq F}$ then ${\displaystyle N\in {\mathcal {F}}{\text{ and }}\mu (N)=0.}$
• Unlike most other properties, this property depends on both ${\displaystyle \operatorname {domain} \mu ={\mathcal {F}}}$ and ${\displaystyle \mu }$'s values.
• 𝜎-finite if there exists a sequence ${\displaystyle F_{1},F_{2},F_{3},\ldots \,}$ in ${\displaystyle {\mathcal {F}}}$ such that ${\displaystyle \mu \left(F_{i}\right)}$ is finite for every index ${\displaystyle i,}$ and also ${\displaystyle \bigcup _{n=1}^{\infty }F_{n}=\bigcup _{F\in {\mathcal {F}}}F.}$

Arbitrary sums

As described in this article, for any family ${\displaystyle \left(r_{i}\right)_{i\in I}}$ of real numbers indexed by an arbitrary indexing set ${\displaystyle I,}$ it is possible to define the sum of the generalized series ${\displaystyle \sum _{i\in I}r_{i},}$ which as expected is denoted by ${\displaystyle \sum _{i\in I}r_{i}}$ (if this limit exists or diverges to ${\displaystyle \pm \infty }$). For example, if ${\displaystyle r_{i}=0}$ for every ${\displaystyle i\in I}$ then ${\displaystyle \sum _{i\in I}r_{i}=0;}$ while if ${\displaystyle I=\mathbb {N} }$ then ${\displaystyle \sum _{i\in I}r_{i}}$ converges in ${\displaystyle \mathbb {R} }$ if and only if ${\displaystyle \sum _{i=1}^{\infty }r_{i}}$ converges unconditionally (or equivalently, converges absolutely). It is known that if a generalized series ${\displaystyle \sum _{i\in I}r_{i}}$ converges in ${\displaystyle \mathbb {R} }$ with its usual Euclidean topology (which implies that both ${\displaystyle \sum _{\stackrel {i\in I}{r_{i}>0}}r_{i}}$ and ${\displaystyle \sum _{\stackrel {i\in I}{r_{i}<0}}r_{i}}$ also converges to elements of ${\displaystyle \mathbb {R} }$) then the set ${\displaystyle \left\{i\in I:r_{i}\neq 0\right\}}$ is necessarily countable (that is, either finite or countably infinite); this remains true if ${\displaystyle \mathbb {R} }$ is replaced with any normed space. From ${\displaystyle \sum _{\stackrel {i\in I,}{r_{i}=0}}r_{i}=0}$ it follows that ${\displaystyle \sum _{i\in I}r_{i}=\sum _{\stackrel {i\in I,}{r_{i}=0}}r_{i}+\sum _{\stackrel {i\in I,}{r_{i}\neq 0}}r_{i}=\sum _{\stackrel {i\in I,}{r_{i}\neq 0}}r_{i}}$ where the series on the right hand side is the sum of a countable set of real numbers. Thus due to the nature of the real numbers and its topology, the definition of "countably additive" is rarely extended from countably many sets ${\displaystyle F_{1},F_{2},\ldots \,}$ in ${\displaystyle {\mathcal {F}}}$ (and the usual countable series ${\displaystyle \sum _{i=1}^{\infty }\mu \left(F_{i}\right)}$) to arbitrarily many sets ${\displaystyle \left(F_{i}\right)_{i\in I}}$ (and the generalized series ${\displaystyle \sum _{i\in I}\mu \left(F_{i}\right)}$).

### Inner measures, outer measures, and other properties

A set function ${\displaystyle \mu }$ is said to be/satisfies[1]

• monotone if ${\displaystyle \mu (E)\leq \mu (F)}$ whenever ${\displaystyle E,F\in {\mathcal {F}}}$ satisfy ${\displaystyle E\subseteq F.}$
• modular if ${\displaystyle \mu (E\cup F)+\mu (E\cap F)=\mu (E)+\mu (F)}$ for all ${\displaystyle E,F\in {\mathcal {F}}}$ such that ${\displaystyle E\cup F,E\cap F\in {\mathcal {F}}.}$
• Every finitely additive function on a field of sets is modular.
• In geometry, real-valued valuations are modular functions.
• submodular if ${\displaystyle \mu (E\cup F)+\mu (E\cap F)\leq \mu (E)+\mu (F)}$ for all ${\displaystyle E,F\in {\mathcal {F}}}$ such that ${\displaystyle E\cup F,E\cap F\in {\mathcal {F}}.}$
• finitely subadditive if ${\displaystyle |\mu (F)|\leq \sum _{i=1}^{n}\left|\mu \left(F_{i}\right)\right|}$ for all finite sequences ${\displaystyle F,F_{1},\ldots ,F_{n}\in {\mathcal {F}}}$ that satisfy ${\displaystyle F\;\subseteq \;\bigcup _{i=1}^{n}F_{i}.}$
• countably subadditive if ${\displaystyle |\mu (F)|\leq \sum _{i=1}^{\infty }\left|\mu \left(F_{i}\right)\right|}$ for all sequences ${\displaystyle F,F_{1},F_{2},F_{3},\ldots \,}$ in ${\displaystyle {\mathcal {F}}}$ that satisfy ${\displaystyle F\;\subseteq \;\bigcup _{i=1}^{\infty }F_{i}.}$
• superadditive if ${\displaystyle \mu (E)+\mu (F)\leq \mu (E\cup F)}$ whenever ${\displaystyle E,F\in {\mathcal {F}}}$ are disjoint with ${\displaystyle E\cup F\in {\mathcal {F}}.}$
• continuous from above if ${\displaystyle \lim _{n\to \infty }\mu \left(F_{i}\right)=\mu \left(\bigcap _{i=1}^{\infty }F_{i}\right)}$ for all non-increasing sequences of sets ${\displaystyle F_{1}\supseteq F_{2}\supseteq F_{3}\cdots \,}$ in ${\displaystyle {\mathcal {F}}}$ such that ${\displaystyle \bigcap _{i=1}^{\infty }F_{i}\in {\mathcal {F}}}$ and ${\displaystyle \mu \left(\bigcap _{i=1}^{\infty }F_{i}\right)}$ is finite.
• continuous from below if ${\displaystyle \lim _{n\to \infty }\mu \left(F_{i}\right)=\mu \left(\bigcup _{i=1}^{\infty }F_{i}\right)}$ for all non-decreasing sequences of sets ${\displaystyle F_{1}\subseteq F_{2}\subseteq F_{3}\cdots \,}$ in ${\displaystyle {\mathcal {F}}}$ such that ${\displaystyle \bigcup _{i=1}^{\infty }F_{i}\in {\mathcal {F}}.}$
• infinity is approached from below if whenever ${\displaystyle F\in {\mathcal {F}}}$ satisfies ${\displaystyle \mu (F)=\infty }$ then for every real ${\displaystyle r>0,}$ there exists some ${\displaystyle F_{r}\in {\mathcal {F}}}$ such that ${\displaystyle F_{r}\subseteq F}$ and ${\displaystyle r\leq \mu \left(F_{r}\right)<\infty .}$
• an outer measure if ${\displaystyle \mu }$ is non-negative, countably subadditive, and has a null empty set.
• an inner measure if ${\displaystyle \mu }$ is non-negative, superadditive, continuous from above, has a null empty set, and ${\displaystyle +\infty }$ is approached from below.

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

• translation invariant if ${\displaystyle \mu (\omega +F)=\mu (F)}$ for all ${\displaystyle \omega \in \Omega }$ and ${\displaystyle F\in {\mathcal {F}}}$ such that ${\displaystyle \omega +F\in {\mathcal {F}}.}$

### Topology related definitions

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

• ${\displaystyle \tau }$-additive if ${\displaystyle \mu \left(\bigcup {\mathcal {D}}\right)=\sup _{D\in {\mathcal {D}}}\mu (D)}$ whenever ${\displaystyle {\mathcal {D}}\subseteq \tau \cap {\mathcal {F}}}$ is directed with respect to ${\displaystyle \,\subseteq \,}$ and satisfies ${\displaystyle \bigcup {\mathcal {D}}:=\bigcup _{D\in {\mathcal {D}}}D\in {\mathcal {F}}.}$
• ${\displaystyle {\mathcal {D}}}$ is directed with respect to ${\displaystyle \,\subseteq \,}$ if and only if it is not empty and for all ${\displaystyle A,B\in {\mathcal {D}}}$ there exists some ${\displaystyle C\in {\mathcal {D}}}$ such that ${\displaystyle A\subseteq C}$ and ${\displaystyle B\subseteq C.}$
• inner regular or tight if for every ${\displaystyle F\in {\mathcal {F}},}$ ${\displaystyle \mu (F)=\sup\{\mu (K):F\supseteq K{\text{ with }}K\in {\mathcal {F}}{\text{ a compact subset of }}(\Omega ,\tau )\}.}$
• outer regular if for every ${\displaystyle F\in {\mathcal {F}},}$ ${\displaystyle \mu (F)=\inf\{\mu (U):F\subseteq U{\text{ and }}U\in {\mathcal {F}}\cap \tau \}.}$
• regular if it is both inner regular and outer regular.
• locally finite if for every point ${\displaystyle \omega \in \Omega }$ there exists some neighborhood ${\displaystyle U\in {\mathcal {F}}\cap \tau }$ of this point such that ${\displaystyle \mu (U)}$ is finite.
• Radon measure if it is a regular and locally finite measure.

### Relationships between set functions

If ${\displaystyle \mu {\text{ and }}\nu }$ are two set functions over ${\displaystyle \Omega ,}$ then:

• ${\displaystyle \mu }$ is said to be absolutely continuous with respect to ${\displaystyle \nu }$ or dominated by ${\displaystyle \nu }$, written ${\displaystyle \mu \ll \nu ,}$ if for every set ${\displaystyle F}$ that belongs to the domain of both ${\displaystyle \mu {\text{ and }}\nu ,}$ if ${\displaystyle \nu (F)=0}$ then ${\displaystyle \mu (F)=0.}$
• If ${\displaystyle \mu }$ and ${\displaystyle \nu }$ are ${\displaystyle \sigma }$-finite measures on the same measurable space and if ${\displaystyle \mu \ll \nu ,}$ then the Radon–Nikodym derivative ${\displaystyle {\frac {d\mu }{d\nu }}}$ exists and for every measurable ${\displaystyle F,}$
${\displaystyle \mu (F)=\int _{F}{\frac {d\mu }{d\nu }}d\nu .}$
• ${\displaystyle \mu {\text{ and }}\nu }$ are singular, written ${\displaystyle \mu \perp \nu ,}$ if there exist disjoint sets ${\displaystyle M}$ and ${\displaystyle N}$ in the domains of ${\displaystyle \mu {\text{ and }}\nu }$ such that ${\displaystyle M\cup N=\Omega ,}$ ${\displaystyle \mu (F)=0}$ for all ${\displaystyle F\subseteq M}$ in the domain of ${\displaystyle \mu ,}$ and ${\displaystyle \nu (F)=0}$ for all ${\displaystyle F\subseteq N}$ in the domain of ${\displaystyle \nu .}$

## Examples

Examples of set functions include:

• The function
${\displaystyle d(A)=\lim _{n\to \infty }{\frac {|A\cap \{1,\ldots ,n\}|}{n}},}$
assigning densities to sufficiently well-behaved subsets ${\displaystyle A\subseteq \{1,2,3,\ldots \},}$ is a set function.
• The Lebesgue measure is a set function that assigns a non-negative real number to any set of real numbers, that is in Lebesgue ${\displaystyle \sigma }$-algebra.[3]
• 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 ${\displaystyle 1,}$ with other sets given probabilities between ${\displaystyle 0}$ and ${\displaystyle 1.}$
• A possibility measure assigns a number between zero and one to each set in the powerset of some given set. See possibility theory.
• A random set is a set-valued random variable. See the article random compact set.

## Properties

### Extending set functions from a semialgebra

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

${\displaystyle \operatorname {algebra} ({\mathcal {F}}):=\left\{F_{1}\sqcup \cdots \sqcup F_{n}:n\in \mathbb {N} {\text{ and }}F_{1},\ldots ,F_{n}\in {\mathcal {F}}{\text{ are pairwise disjoint }}\right\},}$
which is the algebra on ${\displaystyle \Omega }$ generated by ${\displaystyle {\mathcal {F}}.}$ The archetypal example of a semialgebra that is not also an algebra is the family
${\displaystyle {\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}
on ${\displaystyle \Omega :=\mathbb {R} ^{d}}$ where ${\displaystyle (a,b]:=\{x\in \mathbb {R} :a for all ${\displaystyle -\infty \leq a[4] Importantly, the two non-strict inequalities ${\displaystyle \,\leq \,}$ in ${\displaystyle -\infty \leq a_{i} cannot be replaced with strict inequalities ${\displaystyle \,<\,}$ since semialgebras must contain the whole underlying set ${\displaystyle \mathbb {R} ^{d};}$ that is, ${\displaystyle \mathbb {R} ^{d}\in {\mathcal {S}}_{d}}$ is a requirement of semialgebras (as is ${\displaystyle \varnothing \in {\mathcal {S}}_{d}}$).

If ${\displaystyle \mu }$ is finitely additive then it has a unique extension to a set function ${\displaystyle {\overline {\mu }}}$ on ${\displaystyle \operatorname {algebra} ({\mathcal {F}})}$ defined by sending ${\displaystyle F_{1}\sqcup \cdots \sqcup F_{n}\in \operatorname {algebra} ({\mathcal {F}})}$ (so these ${\displaystyle F_{i}\in {\mathcal {F}}}$ are pairwise disjoint) to:[4]

${\displaystyle {\overline {\mu }}\left(F_{1}\sqcup \cdots \sqcup F_{n}\right):=\mu \left(F_{1}\right)+\cdots +\mu \left(F_{n}\right).}$
This extension ${\displaystyle \mu }$ will also be finitely additive: for any pairwise disjoint ${\displaystyle A_{1},\ldots ,A_{n}\in \operatorname {algebra} ({\mathcal {F}}),}$ [4]
${\displaystyle {\overline {\mu }}\left(A_{1}\cup \cdots \cup A_{n}\right)={\overline {\mu }}\left(A_{1}\right)+\cdots +{\overline {\mu }}\left(A_{n}\right).}$

If in addition ${\displaystyle \mu }$ is extended real-valued and monotone (which, in particular, will be the case if ${\displaystyle \mu }$ is non-negative) then ${\displaystyle {\overline {\mu }}}$ will be monotone and finitely subadditive: for any ${\displaystyle A,A_{1},\ldots ,A_{n}\in \operatorname {algebra} ({\mathcal {F}})}$ such that ${\displaystyle A\subseteq A_{1}\cup \cdots \cup A_{n},}$[4]

${\displaystyle {\overline {\mu }}\left(A_{1}\cup \cdots \cup A_{n}\right)\leq {\overline {\mu }}\left(A_{1}\right)+\cdots +{\overline {\mu }}\left(A_{n}\right).}$

## Notes

1. ^ a b Durrett 2019, pp. 1–37, 455–470.
2. ^ Durrett 2019, pp. 466–470.
3. ^ Kolmogorov and Fomin 1975
4. ^ a b c d Durrett 2019, pp. 1–9.

## References

• Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020.
• Kolmogorov, Andrey; Fomin, Sergei V. (1957). Elements of the Theory of Functions and Functional Analysis. Dover Books on Mathematics. New York: Dover Books. ISBN 978-1-61427-304-2. OCLC 912495626.
• A. N. Kolmogorov and S. V. Fomin (1975), Introductory Real Analysis, Dover. ISBN 0-486-61226-0
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.