Set function

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 ${\displaystyle \sum _{i\in I}\mu \left(F_{i}\right)}$).

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 ${\displaystyle \omega +F\in {\mathcal {F}}.}$

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.

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 ${\displaystyle \nu .}$

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

$${\displaystyle {\mathcal {S}}_{d}:=\{\varnothing \}\cup \left\{\left(a_{1},b_{1}\right]\times \cdots \times \left(a_{1},b_{1}\right]~:~-\infty \leq a_{i}<b_{i}\leq \infty {\text{ for all }}i=1,\ldots ,d\right\}}$$

on ${\displaystyle \Omega$ :=\mathbb {R} ^{d}} where ${\displaystyle (a,b]:=\{x\in \mathbb {R} :a<x\leq b\}}$ for all ${\displaystyle -\infty \leq a<b\leq \infty .}$[4] Importantly, the two non-strict inequalities ${\displaystyle \,\leq \,}$ in ${\displaystyle -\infty \leq a_{i}<b_{i}\leq \infty }$ 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]

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