Editing Set function

This article is mirrored from Wikipedia revision 1251642420. Publishing an edit makes a Quire copy: its history starts with that revision, credited to Wikipedia, followed by your change.

You are not logged in. Your edit will be published as anonymous, marked with a short code. Log in or create an account to have edits credited to you.

In mathematics, especially [[measure theory]], a '''set function''' is a [[Function (mathematics)|function]] whose [[Domain of a function|domain]] is a [[Family of sets|family]] of [[Subset|subsets]] of some given set and that (usually) takes its values in the [[extended real number line]] {{mathimg|455f58c7a909e26eeb0260b90f0ff40eb890823a|width=11.364ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; \cup \&#123;\pm \infty \&#125;,&#125;}} which consists of the [[Real number|real numbers]] {{mathimg|786849c765da7a84dbc3cce43e96aad58a5868dc|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; &#125;}} and {{mathimg|3f9a9cde25d110d8d28bb4c482d8d2088f5494c3|width=4.779ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \pm \infty .&#125;}}

A set function generally aims to ''measure'' subsets in some way. [[Measure (mathematics)|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 {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} is a [[family of sets]] over {{mathimg|24b0d5ca6f381068d756f6337c08e0af9d1eeb6f|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \Omega &#125;}} (meaning that {{mathimg|4efe868a8ecbfe0176894f4e6081c8df92a4a941|width=9.991ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F\subseteq \wp (\Omega )&#125;}} where {{mathimg|d598c7f2d38b32ee7f7c4f0ad2fe9f4acfc5a170|width=4.966ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \wp (\Omega )&#125;}} denotes the [[powerset]]) then a ''set function on {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}}'' is a function {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} with [[Domain of a function|domain]] {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} and [[codomain]] {{mathimg|e13233867b861889693a36843d98e51d90d38f9f|width=8.783ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#91;-\infty ,\infty &#93;&#125;}} or, sometimes, the codomain is instead some [[vector space]], as with [[Vector measure|vector measures]], [[Complex measure|complex measures]], and [[Projection-valued measure|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.

{| class="wikitable mw-collapsible mw-collapsed"
|-
!colspan="11" | [[Family of sets|Families {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} of sets]] over {{mathimg|24b0d5ca6f381068d756f6337c08e0af9d1eeb6f|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \Omega &#125;}}
|-
! Is necessarily true of {{mathimg|2c806bc7022198fb7b8ddd4a0b376329bb77e00c|width=2.573ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F\colon &#125;}} or, is {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} [[Closure (mathematics)|closed under:]]
| [[Directed set|Directed by {{mathimg|53cb0e053dc05ebbe0bdad4d92db17770b5ec0b1|width=2.195ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle \,\supseteq &#125;}}]]
| {{mathimg|bb27b38cf9eac6060e67b61f66cd9beec5067f81|width=6.09ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle A\cap B&#125;}}
| {{mathimg|cdb575990bcfbcdf616aa6fd76e8b30bf7fd2169|width=6.09ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle A\cup B&#125;}}
| {{mathimg|4f9fc9ff1d2b0248677aed6da1a89025a859476c|width=5.702ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle B\setminus A&#125;}}
| {{mathimg|8d7c608304f79a1c2259b22557fa42e8f4ad2989|width=5.616ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \Omega \setminus A&#125;}}
| {{mathimg|a3f58c5fa4fae302ac6c8b53024bc635df596616|width=13.483ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle A_&#123;1&#125;\cap A_&#123;2&#125;\cap \cdots &#125;}}
| {{mathimg|1b3f7c4058ec9fab0aabd02f72687a3d92cd6b8a|width=13.483ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle A_&#123;1&#125;\cup A_&#123;2&#125;\cup \cdots &#125;}}
| {{mathimg|e58d6c9cfaccd8b03c9cb1acffb3857023613120|width=6.445ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \Omega \in &#123;\mathcal &#123;F}}
| {{mathimg|70072c0abfe332eb54e47c096c4fbb366ea92911|width=6.575ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \varnothing \in &#123;\mathcal &#123;F}}
| [[Finite intersection property|F.I.P.]]
|-
! [[Pi-system|π-system]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| 
|-
! [[Ring of sets#semiring|Semiring]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| Never
|-
! [[Ring of sets#semialgebra|Semialgebra (semifield)]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| Never
|-
! [[Monotone class]]
| 
| 
| 
| 
| 
| only if {{mathimg|3ba4f0f9c907ac9321bf8494f69cc190cbf8a56d|width=5.512ex|height=2.676ex|va=-0.838ex|alt=&#123;\displaystyle A_&#123;i&#125;\searrow &#125;}}
| only if {{mathimg|b851ff0dcb2264bbedafbef85a71e4f98c842865|width=5.512ex|height=2.676ex|va=-0.671ex|alt=&#123;\displaystyle A_&#123;i&#125;\nearrow &#125;}}
| 
| 
| 
|-
! [[Dynkin system|𝜆-system (Dynkin system)]]
| 
| 
| 
| only if {{mathimg|b09068bd2f7ba899aeb883ebe670b2ad07b0c851|width=6.606ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle A\subseteq B&#125;}}
| 
| 
| only if {{mathimg|b851ff0dcb2264bbedafbef85a71e4f98c842865|width=5.512ex|height=2.676ex|va=-0.671ex|alt=&#123;\displaystyle A_&#123;i&#125;\nearrow &#125;}} or they are [[Disjoint sets|disjoint]]
| 
| 
| Never
|-
! [[Ring of sets|Ring (order theory)]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| 
|-
! [[Ring of sets|Ring (measure theory)]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| Never
|-
! [[Delta-ring|δ-ring]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| Never
|-
! [[Sigma-ring|𝜎-ring]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| Never
|-
! [[Field of sets|Algebra (field)]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| Never
|-
! [[Σ-algebra|𝜎-algebra (𝜎-field)]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| Never
|-
! [[Filter on a set|Filter]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| 
|-
! [[Proper filter on a set|Proper filter]]
| 
| 
| 
| Never
| Never
| 
| 
| 
| Never
| 
|-
! [[Prefilter|Prefilter (filter base)]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| 
|-
! [[Filter subbase]]
| 
| 
| 
| 
| 
| 
| 
| 
| 
| 
|-
! [[Topology (structure)|Open topology]]
| 
| 
| 
| 
| 
| 
| (even arbitrary {{mathimg|e8ff7d0293ad19b43524a133ae5129f3d71f2040|width=1.55ex|height=2.009ex|va=-0.338ex|alt=&#123;\displaystyle \cup &#125;}})
| 
| 
| Never
|-
! [[Topology (structure)|Closed topology]]
| 
| 
| 
| 
| 
| (even arbitrary {{mathimg|9d4e886e6f5a28a33e073fb108440c152ecfe2d3|width=1.55ex|height=2.009ex|va=-0.338ex|alt=&#123;\displaystyle \cap &#125;}})
| 
| 
| 
| Never
|-
! Is necessarily true of {{mathimg|2c806bc7022198fb7b8ddd4a0b376329bb77e00c|width=2.573ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F\colon &#125;}} or, is {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} [[Closure (mathematics)|closed under:]]
| [[Directed downward|directed downward]]
| finite intersections
| finite unions
| relative complements
| complements in {{mathimg|24b0d5ca6f381068d756f6337c08e0af9d1eeb6f|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \Omega &#125;}}
| countable intersections
| countable unions
| contains {{mathimg|24b0d5ca6f381068d756f6337c08e0af9d1eeb6f|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \Omega &#125;}}
| contains {{mathimg|00595c5e33692e724937fdcc8870496acce1ac74|width=1.808ex|height=2.009ex|va=-0.338ex|alt=&#123;\displaystyle \varnothing &#125;}}
| [[Finite intersection property|Finite intersection property]]
|-
|colspan="11" | Additionally, a '''''[[Ring of sets#semiring|semiring]]''''' is a [[Pi-system|π-system]] where every complement {{mathimg|4f9fc9ff1d2b0248677aed6da1a89025a859476c|width=5.702ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle B\setminus A&#125;}} is equal to a finite [[disjoint union]] of sets in {{mathimg|6e1656ae73ede684468b360e948a8a38e6e2c461|width=2.573ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F.&#125;}} A '''''[[Ring of sets#semialgebra|semialgebra]]''''' is a semiring where every complement {{mathimg|8d7c608304f79a1c2259b22557fa42e8f4ad2989|width=5.616ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \Omega \setminus A&#125;}} is equal to a finite [[disjoint union]] of sets in {{mathimg|6e1656ae73ede684468b360e948a8a38e6e2c461|width=2.573ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F.&#125;}} {{mathimg|40c652dd75111bace9e1c2a514a15157f6f8866b|width=15.961ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle A,B,A_&#123;1&#125;,A_&#123;2&#125;,\ldots &#125;}} are arbitrary elements of {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} and it is assumed that {{mathimg|8ed685bdf4c75742b28ccec093cae48329c1a9d6|width=7.48ex|height=2.676ex|va=-0.838ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F\neq \varnothing .&#125;}}
|}

In general, it is typically assumed that {{mathimg|204272c220b816848b35841dd1b5a1e2264c2c0c|width=12.779ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (E)+\mu (F)&#125;}} is always [[well-defined]] for all {{mathimg|18965531bd42b78191c109035e4aa105cca54c16|width=9.964ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle E,F\in &#123;\mathcal &#123;F,&#125;}} or equivalently, that {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} does not take on both {{mathimg|ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1|width=4.132ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle -\infty &#125;}} and {{mathimg|bddbb0e4420a7e744cf71bd71216e11b0bf88831|width=4.132ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle +\infty &#125;}} 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 {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is [[#finitely additive|finitely additive]]:

: ''Set difference formula'': {{mathimg|0457654bf62457e7ff926a73e185632f6805ab01|width=48.239ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (F)-\mu (E)&#61;\mu (F\setminus E)&#123;\text&#123; whenever \mu (F)-\mu (E)&#125;}} is defined with {{mathimg|2ae0e18706cbf389869cdf1c69f6ab9ec3c28c09|width=9.318ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle E,F\in &#123;\mathcal &#123;F}} satisfying {{mathimg|ed17203151e8bf062a1e144c3cfbd9e19421877f|width=6.615ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle E\subseteq F&#125;}} and {{mathimg|33e65bdeac5e8ed24a92796492e36da79387f347|width=11.125ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle F\setminus E\in &#123;\mathcal &#123;F.&#125;}}

'''Null sets'''

A set {{mathimg|2f63bed964e89bbd2f00f55a26858febaa674872|width=6.508ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle F\in &#123;\mathcal &#123;F}} is called a ''null set'' (with respect to {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}}) or simply ''null'' if {{mathimg|83535ef1b0e557e968b577395fb154a14878d8e2|width=9.86ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (F)&#61;0.&#125;}} Whenever {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is not identically equal to either {{mathimg|ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1|width=4.132ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle -\infty &#125;}} or {{mathimg|bddbb0e4420a7e744cf71bd71216e11b0bf88831|width=4.132ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle +\infty &#125;}} then it is typically also assumed that:

* ''null empty set'': {{mathimg|db2123f46749538dc9eceabf5503f6b5c36ed83f|width=9.28ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (\varnothing )&#61;0&#125;}} if {{mathimg|c3f37264659309ae1379415c23f13d833d4cd5b7|width=7.222ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \varnothing \in &#123;\mathcal &#123;F.&#125;}}

'''Variation and mass'''

The [[Total variation (measure theory)|''total variation of a set'']] {{mathimg|4611d85173cd3b508e67077d4a1252c9c05abca2|width=1.499ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle S&#125;}} is {{mathimg|e7e2abe9440b619a52df82dade475d69bd575f31|width=41.125ex|height=3.509ex|va=-0.838ex|alt=&#123;\displaystyle &#124;\mu &#124;(S)~&#123;\stackrel &#123;\scriptscriptstyle &#123;\text&#123;def&#123;&#61;~\sup\&#123;&#124;\mu (F)&#124;:F\in &#123;\mathcal &#123;F&#123;\text&#123; and F\subseteq S\}} where {{mathimg|a9fdfe18b862667a7e7299176aaada2a7e5a9729|width=3.747ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#124;\,\cdot \,&#124;&#125;}} denotes the [[absolute value]] (or more generally, it denotes the [[Norm (mathematics)|norm]] or [[seminorm]] if {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is vector-valued in a ([[Seminormed space|semi]])[[normed space]]). Assuming that {{mathimg|932088cd342a482d92fddb04cab4902342e24978|width=18.065ex|height=5.343ex|va=-2.671ex|alt=&#123;\displaystyle \cup &#123;\mathcal &#123;F~&#123;\stackrel &#123;\scriptscriptstyle &#123;\text&#123;def&#123;&#61;~\textstyle \bigcup \limits _&#123;F\in &#123;\mathcal &#123;FF\in &#123;\mathcal &#123;F,&#125;}} then {{mathimg|8698956ccba7c15d26ae6eb532b0498c76b4b9bd|width=8.369ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#124;\mu &#124;\left(\cup &#123;\mathcal &#123;F\right)&#125;}} is called the ''total variation'' of {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} and {{mathimg|86669ce54d515fe52deab4843f366d55d24dee8f|width=7.075ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu \left(\cup &#123;\mathcal &#123;F\right)&#125;}} is called the ''mass'' of {{mathimg|a1ef6db045c1f6193799bd25a4b68ba9f78646d2|width=2.049ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu .&#125;}}

A set function is called ''finite'' if for every {{mathimg|b79bab9de399c6262d6d4197d8e48134b47d9c9c|width=7.155ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F\in &#123;\mathcal &#123;F,&#125;}} the value {{mathimg|bbcee02992ae0eb8e17cf8a70a03fc2bb5849aea|width=4.952ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (F)&#125;}} is ''finite'' (which by definition means that {{mathimg|5c4b6b3b87f59ef810f54e2551fcb6750cf15768|width=10.374ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (F)\neq \infty &#125;}} and {{mathimg|b24f5e56146b30ee51da70bd6442566664e3a114|width=12.182ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (F)\neq -\infty &#125;}}; an ''infinite value'' is one that is equal to {{mathimg|c26c105004f30c27aa7c2a9c601550a4183b1f21|width=2.324ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \infty &#125;}} or {{mathimg|ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1|width=4.132ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle -\infty &#125;}}). Every finite set function must have a finite [[#mass|mass]].

=== Common properties of set functions ===

A set function {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} on {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} is said to be<ref name="FOOTNOTEDurrett20191–37,_455–470-1">[[#CITEREFDurrett2019|Durrett 2019]], pp.&#160;1–37, 455–470.</ref>

* ''non-negative'' if it is valued in {{mathimg|f850b8f222b718ddb41c8163d1a995e461126c37|width=6.461ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#91;0,\infty &#93;.&#125;}}
* [[Finitely additive set function|''finitely additive'']] if {{mathimg|039c9715f1f08428468e451726949c4a3661a970|width=22.605ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;n&#125;\mu \left(F_&#123;i&#125;\right)&#61;\mu \left(\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;n&#125;F_&#123;i&#125;\right)&#125;}} for all [[pairwise disjoint]] finite sequences {{mathimg|a9bb86a540edc5f5850a52052fd7c473c8d61115|width=15.207ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;1&#125;,\ldots ,F_&#123;n&#125;\in &#123;\mathcal &#123;F}} such that {{mathimg|ca9cd15af9db341700291b95b8a5cd2f9f347070|width=10.763ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;n&#125;F_&#123;i&#125;\in &#123;\mathcal &#123;F.&#125;}}
** If {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} is closed under binary [[Union (set theory)|unions]] then {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is finitely additive if and only if {{mathimg|786c2bd8246d14014fc4c08c3a80fb7479099175|width=25.187ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (E\cup F)&#61;\mu (E)+\mu (F)&#125;}} for all disjoint pairs {{mathimg|77c0f35b3881ff0b4849d58e6265a99ae93d3c07|width=9.964ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle E,F\in &#123;\mathcal &#123;F.&#125;}}
** If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is finitely additive and if {{mathimg|70072c0abfe332eb54e47c096c4fbb366ea92911|width=6.575ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \varnothing \in &#123;\mathcal &#123;F}} then taking {{mathimg|fd5b9540ba810ca5ea7c4c25dbb7030143677b15|width=12.815ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle E:&#61;F:&#61;\varnothing &#125;}} shows that {{mathimg|ee5df2a9f3760bc86ec920f62ee86f750775adf9|width=20.996ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (\varnothing )&#61;\mu (\varnothing )+\mu (\varnothing )&#125;}} which is only possible if {{mathimg|db2123f46749538dc9eceabf5503f6b5c36ed83f|width=9.28ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (\varnothing )&#61;0&#125;}} or {{mathimg|46453746aaf3bea174b1c2613b20a1f74587120d|width=12.896ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (\varnothing )&#61;\pm \infty ,&#125;}} where in the latter case, {{mathimg|212fb73baf28e1ab95485a529e9130c385fc7b32|width=57.504ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (E)&#61;\mu (E\cup \varnothing )&#61;\mu (E)+\mu (\varnothing )&#61;\mu (E)+(\pm \infty )&#61;\pm \infty &#125;}} for every {{mathimg|86edaa69fff2856eaf6991e53e8355a317a0d5d8|width=6.543ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle E\in &#123;\mathcal &#123;F}} (so only the case {{mathimg|db2123f46749538dc9eceabf5503f6b5c36ed83f|width=9.28ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (\varnothing )&#61;0&#125;}} is useful).
* [[Sigma-additive set function|''countably additive'']] or [[Sigma-additive set function|''σ-additive'']]<ref name="FOOTNOTEDurrett2019466–470-2">[[#CITEREFDurrett2019|Durrett 2019]], pp.&#160;466–470.</ref> if in addition to being finitely additive, for all [[pairwise disjoint]] sequences {{mathimg|0a1d4b00942a98a0093390c289c419074cddcbaa|width=10.276ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;1&#125;,F_&#123;2&#125;,\ldots \,&#125;}} in {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} such that {{mathimg|6c2402623f569223c3117c011d8044ce91e33569|width=10.763ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\in &#123;\mathcal &#123;F,&#125;}} all of the following hold:
*# {{mathimg|f75914f75867db6661245e943a8a32071a4ee452|width=22.605ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\mu \left(F_&#123;i&#125;\right)&#61;\mu \left(\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\right)&#125;}}
*#* The series on the left hand side is defined in the usual way as the limit {{mathimg|aabd6c0f4007358a19543e0d5c64f692ed933ad4|width=38.472ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\mu \left(F_&#123;i&#125;\right)~&#123;\stackrel &#123;\scriptscriptstyle &#123;\text&#123;def&#123;&#61;~&#123;\displaystyle \lim _&#123;n\to \infty \mu \left(F_&#123;1&#125;\right)+\cdots +\mu \left(F_&#123;n&#125;}}
*#* As a consequence, if {{mathimg|6027b24c759fcf2ee1d011a7d7c51979dd525c3b|width=10.109ex|height=2.676ex|va=-0.838ex|alt=&#123;\displaystyle \rho :\mathbb &#123;N&#125; \to \mathbb &#123;N&#125; &#125;}} is any [[permutation]]/[[bijection]] then {{mathimg|40386e1ba656c0bf4c2643115b7dd0bb9348b3ff|width=24.477ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\mu \left(F_&#123;i&#125;\right)&#61;\textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\mu \left(F_&#123;\rho (i)&#125;\right);&#125;}} this is because {{mathimg|7473d027f01441d5387093bfd142536f6d17dcfd|width=15.926ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;&#61;\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;\rho (i)}} and applying this condition (a) twice guarantees that both {{mathimg|f75914f75867db6661245e943a8a32071a4ee452|width=22.605ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\mu \left(F_&#123;i&#125;\right)&#61;\mu \left(\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\right)&#125;}} and {{mathimg|3d65228718f93e30af6ee8b94d36cb904a5653c9|width=27.184ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \mu \left(\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;\rho (i)&#125;\right)&#61;\textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\mu \left(F_&#123;\rho (i)&#125;\right)&#125;}} hold. By definition, a convergent series with this property is said to be [[Unconditional convergence|unconditionally convergent]]. Stated in [[plain English]], this means that rearranging/relabeling the sets {{mathimg|c02926e56e1188b93f5ad1b59563a1c98d5ed141|width=9.889ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;1&#125;,F_&#123;2&#125;,\ldots &#125;}} to the new order {{mathimg|40e19a44d55097b519ea72c0cb8bdaed25bf1cd9|width=14.147ex|height=3.009ex|va=-1.171ex|alt=&#123;\displaystyle F_&#123;\rho (1)&#125;,F_&#123;\rho (2)&#125;,\ldots &#125;}} does not affect the sum of their measures. This is desirable since just as the union {{mathimg|ec7275d1aceda8492c24d56012ee3a961e4ef2ca|width=10.629ex|height=5.343ex|va=-2.671ex|alt=&#123;\displaystyle F~&#123;\stackrel &#123;\scriptscriptstyle &#123;\text&#123;def&#123;&#61;~\textstyle \bigcup \limits _&#123;i\in \mathbb &#123;N&#125; &#125;F_&#123;i}} does not depend on the order of these sets, the same should be true of the sums {{mathimg|c23e6e5258faa70db1890183ef5c84a980c35733|width=28.748ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (F)&#61;\mu \left(F_&#123;1&#125;\right)+\mu \left(F_&#123;2&#125;\right)+\cdots &#125;}} and {{mathimg|afa8091db15d1ecf8acda312f81291c3fb515922|width=35.069ex|height=3.343ex|va=-1.171ex|alt=&#123;\displaystyle \mu (F)&#61;\mu \left(F_&#123;\rho (1)&#125;\right)+\mu \left(F_&#123;\rho (2)&#125;\right)+\cdots \,.&#125;}}
*# if {{mathimg|4fe2318e7a9deb2a886d2ff7341722e9c1ee5ae4|width=10.559ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \mu \left(\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\right)&#125;}} is not infinite then this series {{mathimg|c016eeb655a65a127767b5966d9d0daf2f629b5f|width=8.947ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\mu \left(F_&#123;i&#125;\right)&#125;}} must also [[Absolute convergence|converge absolutely]], which by definition means that {{mathimg|8530bee6f0f024bb1e892fbbf6440fc4a296c518|width=10.241ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\left&#124;\mu \left(F_&#123;i&#125;\right)\right&#124;&#125;}} must be finite. This is automatically true if {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is [[#non-negative|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 {{mathimg|c9b2de288322540ab825257592d5f849b6b0600d|width=47.114ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\mu \left(F_&#123;i&#125;\right)&#61;&#123;\displaystyle \lim _&#123;N\to \infty \mu \left(F_&#123;1&#125;\right)+\mu \left(F_&#123;2&#125;\right)+\cdots +\mu \left(F_&#123;N&#125;\right)&#125;}} 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 {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is valued in {{mathimg|e7c53f28e818adb1f7baadb9f66eb45aebc1c33d|width=9.43ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#91;-\infty ,\infty &#93;.&#125;}}
*# if {{mathimg|24076e95b0143fe342e94d8008e4dc37e1a65a93|width=22.605ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \mu \left(\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\right)&#61;\textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\mu \left(F_&#123;i&#125;\right)&#125;}} is infinite then it is also required that the value of at least one of the series {{mathimg|e29255ef7794e15ddbe3161205a512ed84104a61|width=32.283ex|height=6.509ex|va=-4.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;\stackrel &#123;i\in \mathbb &#123;N&#125; &#125;&#123;\mu \left(F_&#123;i&#125;\right)>0\mu \left(F_&#123;i&#125;\right)\;&#123;\text&#123; and \;\textstyle \sum \limits _&#123;\stackrel &#123;i\in \mathbb &#123;N&#125; &#125;&#123;\mu \left(F_&#12}} be finite (so that the sum of their values is well-defined). This is automatically true if {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is [[#non-negative|non-negative]].
* a [[Pre-measure|''pre-measure'']] if it is [[#non-negative|non-negative]], [[Sigma-additive set function|countably additive]] (including [[#Finitely additive|finitely additive]]), and has a [[#null empty set|null empty set]].
* a [[Measure (mathematics)|''measure'']] if it is a [[#pre-measure|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|null empty set]].
* a [[Probability measure|''probability measure'']] if it is a measure that has a [[#mass|mass]] of {{mathimg|af8c4e445819b13a052647aa3eb2be990b0a4b24|width=1.809ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle 1.&#125;}}
* an [[Outer measure|''outer measure'']] if it is non-negative, [[#countably subadditive|countably subadditive]], has a [[#null empty set|null empty set]], and has the [[power set]] {{mathimg|d598c7f2d38b32ee7f7c4f0ad2fe9f4acfc5a170|width=4.966ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \wp (\Omega )&#125;}} as its domain.
** Outer measures appear in the [[Carathéodory's extension theorem]] and they are often [[Restriction (mathematics)|restricted]] to [[Carathéodory's criterion|Carathéodory measurable subsets]]
* a [[Signed measure|''signed measure'']] if it is countably additive, has a [[#null empty set|null empty set]], and {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} does not take on both {{mathimg|ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1|width=4.132ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle -\infty &#125;}} and {{mathimg|bddbb0e4420a7e744cf71bd71216e11b0bf88831|width=4.132ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle +\infty &#125;}} as values.
* [[Complete measure|''complete'']] if every subset of every [[#null set|null set]] is null; explicitly, this means: whenever {{mathimg|6c14ecc94dd792510ebe06754706e05b017578cf|width=24.736ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle F\in &#123;\mathcal &#123;F&#123;\text&#123; satisfies \mu (F)&#61;0&#125;}} and {{mathimg|1f797a8bcc68d8c2dd485ef4710cd73c630a8e36|width=6.903ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle N\subseteq F&#125;}} is any subset of {{mathimg|545fd099af8541605f7ee55f08225526be88ce57|width=1.741ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle F&#125;}} then {{mathimg|e259bb2da2217318fbe5674f130f437444efb4e9|width=6.831ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle N\in &#123;\mathcal &#123;F}} and {{mathimg|05f43bc897c2efd70791476dbb676abeccc16ddc|width=10.182ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (N)&#61;0.&#125;}}
** Unlike many other properties, completeness places requirements on the set {{mathimg|90dbb3965efea1a713b0cf1a9877b0335f5b9082|width=14.307ex|height=2.676ex|va=-0.838ex|alt=&#123;\displaystyle \operatorname &#123;domain&#125; \mu &#61;&#123;\mathcal &#123;F}} (and not just on {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}}'s values).
* [[Σ-finite measure|''𝜎-finite'']] if there exists a sequence {{mathimg|63061cc12b611b5d2f78618233e707ee7d008f59|width=13.859ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;1&#125;,F_&#123;2&#125;,F_&#123;3&#125;,\ldots \,&#125;}} in {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} such that {{mathimg|e64c5f07d88dfcbe5ff415cd220fc9491bcdf1d1|width=5.892ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu \left(F_&#123;i&#125;\right)&#125;}} is finite for every index {{mathimg|d3d0f7dadba3056fa3c06a6bee5c0b4182471152|width=1.449ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle i,&#125;}} and also {{mathimg|7f6070d189e049746ee06a9b092cd804ea885da0|width=15.75ex|height=6.176ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \bigcup \limits _&#123;n&#61;1&#125;^&#123;\infty &#125;F_&#123;n&#125;&#61;\textstyle \bigcup \limits _&#123;F\in &#123;\mathcal &#123;FF.&#125;}}
* [[Decomposable measure|''decomposable'']] if there exists a subfamily {{mathimg|1d2ffbbf160c6de18a1844fdcd98f72dc78a90f2|width=6.729ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle &#123;\mathcal &#123;P\subseteq &#123;\mathcal &#123;F}} of pairwise disjoint sets such that {{mathimg|b9dc683a0d2c231cc914c3ec101994ef35d8c53b|width=4.956ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (P)&#125;}} is finite for every {{mathimg|bb727fbfb3d66abc8c9a7e7b649323f87b85a98d|width=6.29ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle P\in &#123;\mathcal &#123;P}} and also {{mathimg|b573c31c7bd8d2ad14c3507e2d7a7c402e9e6647|width=14.971ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \bigcup \limits _&#123;P\in &#123;\mathcal &#123;P\,P&#61;\textstyle \bigcup \limits _&#123;F\in &#123;\mathcal &#123;FF&#125;}} (where {{mathimg|313b47b23fd6eea800abb5767facf6c4f8eeccfa|width=14.307ex|height=2.676ex|va=-0.838ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F&#61;\operatorname &#123;domain&#125; \mu &#125;}}).
** Every 𝜎-finite set function is decomposable although not conversely. For example, the [[counting measure]] on {{mathimg|786849c765da7a84dbc3cce43e96aad58a5868dc|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; &#125;}} (whose domain is {{mathimg|6271848da40d4b8be111f20bd3381bdaeaaa925d|width=4.966ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \wp (\mathbb &#123;R&#125; )&#125;}}) is decomposable but not 𝜎-finite.
* a [[Vector measure|''vector measure'']] if it is a countably additive set function {{mathimg|21080970c7d2d269bd3842aa4f4bdc2a63571659|width=10.86ex|height=2.676ex|va=-0.838ex|alt=&#123;\displaystyle \mu :&#123;\mathcal &#123;F\to X&#125;}} valued in a [[topological vector space]] {{mathimg|68baa052181f707c662844a465bfeeb135e82bab|width=1.98ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle X&#125;}} (such as a [[normed space]]) whose domain is a [[σ-algebra]].
** If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is valued in a [[normed space]] {{mathimg|258512fa727a69eec5255c3049ca4f999abd74ab|width=8.827ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle (X,\&#124;\cdot \&#124;)&#125;}} then it is countably additive if and only if for any [[pairwise disjoint]] sequence {{mathimg|0a1d4b00942a98a0093390c289c419074cddcbaa|width=10.276ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;1&#125;,F_&#123;2&#125;,\ldots \,&#125;}} in {{mathimg|9eaedf8742be7f4e854f921b2ae3adcd1ad041da|width=2.573ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F,&#125;}} {{mathimg|9facd4446aa9f4f78ede309445fdaeff459b2e67|width=46.154ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \lim _&#123;n\to \infty &#125;\left\&#124;\mu \left(F_&#123;1&#125;\right)+\cdots +\mu \left(F_&#123;n&#125;\right)-\mu \left(\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\right)\right\&#124;&#61;0.&#125;}} If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is finitely additive and valued in a [[Banach space]] then it is countably additive if and only if for any pairwise disjoint sequence {{mathimg|0a1d4b00942a98a0093390c289c419074cddcbaa|width=10.276ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;1&#125;,F_&#123;2&#125;,\ldots \,&#125;}} in {{mathimg|9eaedf8742be7f4e854f921b2ae3adcd1ad041da|width=2.573ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F,&#125;}} {{mathimg|990ad5d4a0f72c6d4448b1d37f53271d6df12320|width=38.302ex|height=3.843ex|va=-1.838ex|alt=&#123;\displaystyle \lim _&#123;n\to \infty &#125;\left\&#124;\mu \left(F_&#123;n&#125;\cup F_&#123;n+1&#125;\cup F_&#123;n+2&#125;\cup \cdots \right)\right\&#124;&#61;0.&#125;}}
* a [[Complex measure|''complex measure'']] if it is a countably additive [[Complex number|complex]]-valued set function {{mathimg|eff276aab36586af33a0d4a3e9298898702de148|width=10.558ex|height=2.676ex|va=-0.838ex|alt=&#123;\displaystyle \mu :&#123;\mathcal &#123;F\to \mathbb &#123;C&#125; &#125;}} whose domain is a [[σ-algebra]].
** By definition, a complex measure never takes {{mathimg|c586ae37f8efec026b8a4ea3f6a5253576c2c4e6|width=4.132ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \pm \infty &#125;}} as a value and so has a [[#null empty set|null empty set]].
* a [[Random measure|''random measure'']] if it is a measure-valued [[random element]].

'''Arbitrary sums'''

As described [[Series (mathematics)#Summations over arbitrary index sets|in this article's section on generalized series]], for any family {{mathimg|98e0fc44f5697462fbfd34ad5321cab8c479af55|width=6.382ex|height=3.009ex|va=-1.005ex|alt=&#123;\displaystyle \left(r_&#123;i&#125;\right)_&#123;i\in I}} of [[Real number|real numbers]] indexed by an arbitrary [[indexing set]] {{mathimg|a12504e3a6f191d6fb24fb4a6795266bdd171664|width=1.819ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle I,&#125;}} it is possible to define their sum {{mathimg|7bbdbfb3fba6aee58e5f96edc92d813dc28bed0a|width=4.728ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in I&#125;r_&#123;i}} as the limit of the [[Net (mathematics)|net]] of finite partial sums {{mathimg|daba298850e53d76b41f0861edb4a28df0948b17|width=29.995ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle F\in \operatorname &#123;FiniteSubsets&#125; (I)\mapsto \textstyle \sum \limits _&#123;i\in F&#125;r_&#123;i}} where the domain {{mathimg|6325c076e5aa51acd515eaa01670d7196f25f30d|width=16.669ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \operatorname &#123;FiniteSubsets&#125; (I)&#125;}} is [[Directed set|directed]] by {{mathimg|dc8bb29ab88d5da327b69abf1889c38400947d73|width=3.229ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle \,\subseteq .\,&#125;}} Whenever this [[Convergent net|net converges]] then its limit is denoted by the symbols {{mathimg|7bbdbfb3fba6aee58e5f96edc92d813dc28bed0a|width=4.728ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in I&#125;r_&#123;i}} while if this net instead diverges to {{mathimg|c586ae37f8efec026b8a4ea3f6a5253576c2c4e6|width=4.132ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \pm \infty &#125;}} then this may be indicated by writing {{mathimg|2549037d48551f872b0c8f5b36b97fe9cfb5de02|width=12.605ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in I&#125;r_&#123;i&#125;&#61;\pm \infty .&#125;}} Any sum over the empty set is defined to be zero; that is, if {{mathimg|8df9cea76c8946b9f1b80b951c45a4ff6fda0fd6|width=6.078ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle I&#61;\varnothing &#125;}} then {{mathimg|186726beaaa50e281e47c7472baaa83b82f6ac43|width=9.439ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in \varnothing &#125;r_&#123;i&#125;&#61;0&#125;}} by definition.

For example, if {{mathimg|7dff80ac0fa092ab8f667000ba24625169d3ea26|width=6.142ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle z_&#123;i&#125;&#61;0&#125;}} for every {{mathimg|2d740fe587228ce31b71c9628e089d1a9b37c6be|width=4.815ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle i\in I&#125;}} then {{mathimg|dd5711a2072b9b3e439676f1b92eb686a1c7e0b5|width=9.668ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in I&#125;z_&#123;i&#125;&#61;0.&#125;}} And it can be shown that {{mathimg|7cc91db536223d76c2fa51189fc1ca1ac9586353|width=44.323ex|height=6.509ex|va=-4.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in I&#125;r_&#123;i&#125;&#61;\textstyle \sum \limits _&#123;\stackrel &#123;i\in I,&#125;&#123;r_&#123;i&#125;&#61;0r_&#123;i&#125;+\textstyle \sum \limits _&#123;\stackrel &#123;i\in I,&#125;&#123;r_&#123;i&#125;\neq 0r_&#123;i&#125;&#61;0+\text}} If {{mathimg|f00dfa40411dcce619a2d831188111085817a695|width=5.948ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle I&#61;\mathbb &#123;N&#125; &#125;}} then the generalized series {{mathimg|7bbdbfb3fba6aee58e5f96edc92d813dc28bed0a|width=4.728ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in I&#125;r_&#123;i}} converges in {{mathimg|786849c765da7a84dbc3cce43e96aad58a5868dc|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; &#125;}} if and only if {{mathimg|8829e689dd0b563c5e60e15584519a371494fc07|width=4.903ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;r_&#123;i}} [[Unconditional convergence|converges unconditionally]] (or equivalently, [[Absolute convergence|converges absolutely]]) in the usual sense. If a generalized series {{mathimg|7bbdbfb3fba6aee58e5f96edc92d813dc28bed0a|width=4.728ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in I&#125;r_&#123;i}} converges in {{mathimg|786849c765da7a84dbc3cce43e96aad58a5868dc|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; &#125;}} then both {{mathimg|8e9acd8a3e8a7457bc8b6c0f99575a0dee17fc7f|width=5.702ex|height=6.343ex|va=-4.338ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;\stackrel &#123;i\in I&#125;&#123;r_&#123;i&#125;>0r_&#123;i}} and {{mathimg|597d661ac05b0ecaead4c6dc1ec48281c8f927f3|width=5.702ex|height=6.343ex|va=-4.338ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;\stackrel &#123;i\in I&#125;&#123;r_&#123;i&#125;&lt;0r_&#123;i}} also converge to elements of {{mathimg|786849c765da7a84dbc3cce43e96aad58a5868dc|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; &#125;}} and the set {{mathimg|607186e818026fdc20cd7af8bba61776a7917579|width=15.186ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \left\&#123;i\in I:r_&#123;i&#125;\neq 0\right\}} is necessarily [[Countable set|countable]] (that is, either finite or [[countably infinite]]); [[Series (mathematics)#Abelian topological groups|this remains true]] if {{mathimg|786849c765da7a84dbc3cce43e96aad58a5868dc|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; &#125;}} is replaced with any [[normed space]].<ref name="ProofCountablyManyNon0Terms-3">Suppose the net {{mathimg|425988094250c3c30ebf37d4de6a097fa29ec2f9|width=55.349ex|height=6.343ex|va=-2.671ex|alt=&#123;\textstyle \textstyle \sum \limits _&#123;i\in I&#125;r_&#123;i&#125;~&#123;\stackrel &#123;\scriptscriptstyle &#123;\text&#123;def&#123;&#61;~&#123;\textstyle \lim \limits _&#123;A\in \operatorname &#123;Finite&#125; (I)\ \textstyle \sum \limits _&#123;i\in A&#125;r_&#123;i&#125;&#61;\lim \le}} converges to some point in a [[metrizable topological vector space]] {{mathimg|68baa052181f707c662844a465bfeeb135e82bab|width=1.98ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle X&#125;}} (such as {{mathimg|0522388d36b55de7babe4bbfc49475eaf590c2bd|width=2.325ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; ,&#125;}} {{mathimg|c6ff6a3dc2982018ff20f1d2c927afc74a217be6|width=2.325ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle \mathbb &#123;C&#125; ,&#125;}} or a [[normed space]]), where recall that this net's domain is the [[directed set]] {{mathimg|26adb9e4a960a55fda9f109fbfde1835fbd65bb1|width=14.32ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle (\operatorname &#123;Finite&#125; (I),\subseteq ).&#125;}} Like every convergent net, this convergent net of partial sums {{mathimg|a6aae1efd10ff8347a9e9f9c80fe191ec328dde1|width=10.489ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle A\mapsto \textstyle \sum \limits _&#123;i\in A&#125;r_&#123;i}} is a ''[[Cauchy net]]'', which for this particular net means (by definition) that for every neighborhood {{mathimg|54a9c4c547f4d6111f81946cad242b18298d70b7|width=2.435ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle W&#125;}} of the origin in {{mathimg|09ba32eeb405f7f5f2bac1eb12987c47d2fd42df|width=2.627ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle X,&#125;}} there exists a finite subset {{mathimg|7c98dbd929292d464de6942e95db306b8b8a6fca|width=2.797ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle A_&#123;0}} of {{mathimg|535ea7fc4134a31cbe2251d9d3511374bc41be9f|width=1.172ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle I&#125;}} such that {{mathimg|af86a3c9acc8e33b9d3cde738a52995e10a2ca82|width=18.411ex|height=4.676ex|va=-2.671ex|alt=&#123;\textstyle \textstyle \sum \limits _&#123;i\in B&#125;r_&#123;i&#125;-\textstyle \sum \limits _&#123;i\in C&#125;r_&#123;i&#125;\in W&#125;}} for all finite supersets {{mathimg|b1ac1912e11befcec0f9f12401acaa6658a07554|width=11.107ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle B,C\supseteq A_&#123;0&#125;;&#125;}} this implies that {{mathimg|8f57f21052542909e1ed0ace2c348af75565834f|width=7.124ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle r_&#123;i&#125;\in W&#125;}} for every {{mathimg|13b67b5b4b0eb06f665752eb51c1d2f5fb79cee7|width=9.807ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle i\in I\setminus A_&#123;0}} (by taking {{mathimg|b3e9562f5e431530dbbbcb464ede8c3d947d98f3|width=14.017ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle B:&#61;A_&#123;0&#125;\cup \&#123;i\}} and {{mathimg|4a0b2bf964d14fae482459db3f892b0e9519557a|width=8.309ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle C:&#61;A_&#123;0}}). Since {{mathimg|68baa052181f707c662844a465bfeeb135e82bab|width=1.98ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle X&#125;}} is metrizable, it has a countable neighborhood basis {{mathimg|e6b87fd8b17c2bc22bce8b49f0e969881f858ed6|width=10.075ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle U_&#123;1&#125;,U_&#123;2&#125;,\ldots &#125;}} at the origin, whose intersection is necessarily {{mathimg|836ac01e056f2d3ee353991e9c4a172aa767cf45|width=19.758ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle U_&#123;1&#125;\cap U_&#123;2&#125;\cap \cdots &#61;\&#123;0\}} (since {{mathimg|68baa052181f707c662844a465bfeeb135e82bab|width=1.98ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle X&#125;}} is a Hausdorff TVS). For every positive integer {{mathimg|5b98e213fe7ef48da0be47453bc1bb66f37f4eec|width=6.56ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle n\in \mathbb &#123;N&#125; ,&#125;}} pick a finite subset {{mathimg|b594928ef005ae9d280ab3954568a2e8c43c518b|width=7.232ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle A_&#123;n&#125;\subseteq I&#125;}} such that {{mathimg|89d850b85dd29ead1e17515d51795e47a1e3b55e|width=7.495ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle r_&#123;i&#125;\in U_&#123;n}} for every {{mathimg|47d27b848987e61b0972d6086555cd1743ec9fea|width=10.618ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle i\in I\setminus A_&#123;n&#125;.&#125;}} If {{mathimg|add78d8608ad86e54951b8c8bd6c8d8416533d20|width=0.802ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle i&#125;}} belongs to {{mathimg|e4d20dfc23c24ef097e490e648019d6f50c8c353|width=45.592ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle (I\setminus A_&#123;1&#125;)\cap (I\setminus A_&#123;2&#125;)\cap \cdots &#61;I\setminus \left(A_&#123;1&#125;\cup A_&#123;2&#125;\cup \cdots \right)&#125;}} then {{mathimg|a0b6d651eaf432dbf1f106021c8bb499ae83fd1f|width=1.848ex|height=2.009ex|va=-0.671ex|alt=&#123;\displaystyle r_&#123;i}} belongs to {{mathimg|cdd2c9efd95efc829017d4d9b0d073aa44998160|width=20.405ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle U_&#123;1&#125;\cap U_&#123;2&#125;\cap \cdots &#61;\&#123;0\&#125;.&#125;}} Thus {{mathimg|fa1284f602f573939ac65142dfc66bf2763a1a0c|width=6.109ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle r_&#123;i&#125;&#61;0&#125;}} for every index {{mathimg|2d740fe587228ce31b71c9628e089d1a9b37c6be|width=4.815ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle i\in I&#125;}} that does not belong to the countable set {{mathimg|66124c2b164021c5e4d5e30298ff0c15ec3c5db8|width=14.517ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle A_&#123;1&#125;\cup A_&#123;2&#125;\cup \cdots .&#125;}} {{mathimg|8733090f2d787d03101c3e16dc3f6404f0e7dd4c|width=1.808ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \blacksquare &#125;}}</ref> It follows that in order for a generalized series {{mathimg|7bbdbfb3fba6aee58e5f96edc92d813dc28bed0a|width=4.728ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in I&#125;r_&#123;i}} to converge in {{mathimg|786849c765da7a84dbc3cce43e96aad58a5868dc|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; &#125;}} or {{mathimg|c6ff6a3dc2982018ff20f1d2c927afc74a217be6|width=2.325ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle \mathbb &#123;C&#125; ,&#125;}} it is necessary that all but at most countably many {{mathimg|a0b6d651eaf432dbf1f106021c8bb499ae83fd1f|width=1.848ex|height=2.009ex|va=-0.671ex|alt=&#123;\displaystyle r_&#123;i}} will be equal to {{mathimg|95547343453ea34a314dd174f8458012f5a39ca3|width=1.809ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle 0,&#125;}} which means that {{mathimg|cbe8655e5c2122d3fcf6f980962e42c4922cd5c7|width=15.077ex|height=6.343ex|va=-4.338ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in I&#125;r_&#123;i&#125;~&#61;~\textstyle \sum \limits _&#123;\stackrel &#123;i\in I&#125;&#123;r_&#123;i&#125;\neq 0r_&#123;i}} is a sum of at most countably many non-zero terms. Said differently, if {{mathimg|607186e818026fdc20cd7af8bba61776a7917579|width=15.186ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \left\&#123;i\in I:r_&#123;i&#125;\neq 0\right\}} is uncountable then the generalized series {{mathimg|7bbdbfb3fba6aee58e5f96edc92d813dc28bed0a|width=4.728ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in I&#125;r_&#123;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|countably additive]]" is rarely extended from countably many sets {{mathimg|0a1d4b00942a98a0093390c289c419074cddcbaa|width=10.276ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;1&#125;,F_&#123;2&#125;,\ldots \,&#125;}} in {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} (and the usual countable series {{mathimg|c016eeb655a65a127767b5966d9d0daf2f629b5f|width=8.947ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\mu \left(F_&#123;i&#125;\right)&#125;}}) to arbitrarily many sets {{mathimg|7b89c0ece2e279d4ced2ea94a0ee17125fdeed57|width=6.828ex|height=3.009ex|va=-1.005ex|alt=&#123;\displaystyle \left(F_&#123;i&#125;\right)_&#123;i\in I}} (and the generalized series {{mathimg|fdb5daa81dbacca0b0829a8d41ec6f9abb58da81|width=8.772ex|height=4.676ex|va=-2.671ex|alt=&#123;\displaystyle \textstyle \sum \limits _&#123;i\in I&#125;\mu \left(F_&#123;i&#125;\right)&#125;}}).

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

A set function {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is said to be/satisfies<ref name="FOOTNOTEDurrett20191–37,_455–470-1"/>

* ''monotone'' if {{mathimg|74acc89443a10c335d3444a1671b94948437cbb3|width=13.037ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (E)\leq \mu (F)&#125;}} whenever {{mathimg|2ae0e18706cbf389869cdf1c69f6ab9ec3c28c09|width=9.318ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle E,F\in &#123;\mathcal &#123;F}} satisfy {{mathimg|c43b37bd26bea5a94a90bdfa416a873b8574c36d|width=7.262ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle E\subseteq F.&#125;}}
* [[Modular set function|''modular'']] if it satisfies the following condition, known as ''modularity'': {{mathimg|b51cc6bb3e0844ad2cbce1944a5092e90e3c056d|width=37.338ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (E\cup F)+\mu (E\cap F)&#61;\mu (E)+\mu (F)&#125;}} for all {{mathimg|2ae0e18706cbf389869cdf1c69f6ab9ec3c28c09|width=9.318ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle E,F\in &#123;\mathcal &#123;F}} such that {{mathimg|c866026b69b87a6264f454ce07ce44adb78363e8|width=18.646ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle E\cup F,E\cap F\in &#123;\mathcal &#123;F.&#125;}}
** 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)|''valuation'']]. This [[Valuation (geometry)|geometric definition of "valuation"]] should not be confused with the stronger non-equivalent [[Valuation (measure theory)|measure theoretic definition of "valuation"]] that is [[#valuation|given below]].
* [[Submodular set function|''submodular'']] if {{mathimg|627da55962f15659ce2be3ce3fc862ddba0c6c9b|width=37.338ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (E\cup F)+\mu (E\cap F)\leq \mu (E)+\mu (F)&#125;}} for all {{mathimg|2ae0e18706cbf389869cdf1c69f6ab9ec3c28c09|width=9.318ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle E,F\in &#123;\mathcal &#123;F}} such that {{mathimg|c866026b69b87a6264f454ce07ce44adb78363e8|width=18.646ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle E\cup F,E\cap F\in &#123;\mathcal &#123;F.&#125;}}
* ''finitely subadditive'' if {{mathimg|4ea6b1df7d6dfeeb621b5d7c93ed80d098173157|width=19.585ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle &#124;\mu (F)&#124;\leq \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;n&#125;\left&#124;\mu \left(F_&#123;i&#125;\right)\right&#124;&#125;}} for all finite sequences {{mathimg|b230ffab6442c3eeda9b7b9a69a7a32b8584e6a8|width=17.982ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F,F_&#123;1&#125;,\ldots ,F_&#123;n&#125;\in &#123;\mathcal &#123;F}} that satisfy {{mathimg|126273370d5c09dae346e31acae286cf9e1c370a|width=12.126ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle F\;\subseteq \;\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;n&#125;F_&#123;i&#125;.&#125;}}
* ''countably subadditive'' or ''σ-subadditive'' if {{mathimg|944b1b85966e29e01fe1773a431dfe8f995c7340|width=19.585ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle &#124;\mu (F)&#124;\leq \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\left&#124;\mu \left(F_&#123;i&#125;\right)\right&#124;&#125;}} for all sequences {{mathimg|49e76ef548e076ea4c01559d2b1cf37131762924|width=16.633ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F,F_&#123;1&#125;,F_&#123;2&#125;,F_&#123;3&#125;,\ldots \,&#125;}} in {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} that satisfy {{mathimg|783a357f1c2ed8e86876b697d97e000522c4849b|width=12.126ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle F\;\subseteq \;\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;.&#125;}}
** If {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} is closed under finite unions then this condition holds if and only if {{mathimg|156c54c599f98fb67b4c02ed34e61c7d0425670b|width=29.17ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#124;\mu (F\cup G)&#124;\leq &#124;\mu (F)&#124;+&#124;\mu (G)&#124;&#125;}} for all {{mathimg|12270b818a5336ae550c3ca9cb8d721d726f1d6f|width=10.016ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F,G\in &#123;\mathcal &#123;F.&#125;}} If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is non-negative then the absolute values may be removed.
** If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is a measure then this condition holds if and only if {{mathimg|d88f4cf9cb3798a579a41bdd449eacb26fa4f818|width=22.605ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \mu \left(\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\right)\leq \textstyle \sum \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;\mu \left(F_&#123;i&#125;\right)&#125;}} for all {{mathimg|63061cc12b611b5d2f78618233e707ee7d008f59|width=13.859ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;1&#125;,F_&#123;2&#125;,F_&#123;3&#125;,\ldots \,&#125;}} in {{mathimg|6e1656ae73ede684468b360e948a8a38e6e2c461|width=2.573ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F.&#125;}}<ref name="FOOTNOTERoydenFitzpatrick201030-4">[[#CITEREFRoydenFitzpatrick2010|Royden &amp; Fitzpatrick 2010]], p.&#160;30.</ref> If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is a [[probability measure]] then this inequality is [[Boole's inequality]].
** If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is countably subadditive and {{mathimg|70072c0abfe332eb54e47c096c4fbb366ea92911|width=6.575ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \varnothing \in &#123;\mathcal &#123;F}} with {{mathimg|db2123f46749538dc9eceabf5503f6b5c36ed83f|width=9.28ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (\varnothing )&#61;0&#125;}} then {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is [[#finitely subadditive|finitely subadditive]].
* [[Superadditivity|''superadditive'']] if {{mathimg|0871563d7e0f6b994533fea4423028eae8f8cb57|width=25.187ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (E)+\mu (F)\leq \mu (E\cup F)&#125;}} whenever {{mathimg|2ae0e18706cbf389869cdf1c69f6ab9ec3c28c09|width=9.318ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle E,F\in &#123;\mathcal &#123;F}} are disjoint with {{mathimg|e835d543ba4263053be0c9bdb1b44ce6d05b0106|width=11.513ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle E\cup F\in &#123;\mathcal &#123;F.&#125;}}
* ''continuous from above'' if {{mathimg|c3290cda4be22327e69e0f65f508ad9cb6bda5f3|width=24.21ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \lim _&#123;n\to \infty &#125;\mu \left(F_&#123;i&#125;\right)&#61;\mu \left(\textstyle \bigcap \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\right)&#125;}} for all ''non-increasing sequences'' of sets {{mathimg|0999214b445335a1020256c0ac7aead2c6836d34|width=17.341ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;1&#125;\supseteq F_&#123;2&#125;\supseteq F_&#123;3&#125;\cdots \,&#125;}} in {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} such that {{mathimg|6dea6a5a1c57094e66aefa6878056327d540968d|width=10.117ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \bigcap \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\in &#123;\mathcal &#123;F}} with {{mathimg|f2836057f0fb1f1d3b086c3484017b24f2359dc9|width=10.559ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \mu \left(\textstyle \bigcap \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\right)&#125;}} and all {{mathimg|e64c5f07d88dfcbe5ff415cd220fc9491bcdf1d1|width=5.892ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu \left(F_&#123;i&#125;\right)&#125;}} finite.
** Lebesgue measure {{mathimg|b43d0ea3c9c025af1be9128e62a18fa74bedda2a|width=1.355ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \lambda &#125;}} is continuous from above but it would not be if the assumption that all {{mathimg|e64c5f07d88dfcbe5ff415cd220fc9491bcdf1d1|width=5.892ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu \left(F_&#123;i&#125;\right)&#125;}} are eventually finite was omitted from the definition, as this example shows: For every integer {{mathimg|d3d0f7dadba3056fa3c06a6bee5c0b4182471152|width=1.449ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle i,&#125;}} let {{mathimg|18f9a5ed62f131c8ad9acdf7f8539f103f1e0ec9|width=2.294ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;i}} be the open interval {{mathimg|264ac22be7b70d4fcaa3ca3e8c0d1db74ab70bb5|width=5.969ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle (i,\infty )&#125;}} so that {{mathimg|82aa14ad331047101de7c0b2c48c26d840e5ad6e|width=51.953ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \lim _&#123;n\to \infty &#125;\lambda \left(F_&#123;i&#125;\right)&#61;\lim _&#123;n\to \infty &#125;\infty &#61;\infty \neq 0&#61;\lambda (\varnothing )&#61;\lambda \left(\textstyle \bigcap \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\right)&#125;}} where {{mathimg|5465ac68fdca84573d99e970eedcdef44a813ea0|width=10.903ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \bigcap \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;&#61;\varnothing .&#125;}}
* ''continuous from below'' if {{mathimg|e8292bd0968cd1a002956808b5c40eac2e085290|width=24.21ex|height=6.176ex|va=-2.505ex|alt=&#123;\displaystyle \lim _&#123;n\to \infty &#125;\mu \left(F_&#123;i&#125;\right)&#61;\mu \left(\textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\right)&#125;}} for all ''non-decreasing sequences'' of sets {{mathimg|78c4e25055e8ef156d61d9a524d4b8b64268a6c8|width=17.341ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;1&#125;\subseteq F_&#123;2&#125;\subseteq F_&#123;3&#125;\cdots \,&#125;}} in {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} such that {{mathimg|6b91226cc8688fcedc55aa8c65ba9aff82ef355b|width=10.763ex|height=6.009ex|va=-2.505ex|alt=&#123;\displaystyle \textstyle \bigcup \limits _&#123;i&#61;1&#125;^&#123;\infty &#125;F_&#123;i&#125;\in &#123;\mathcal &#123;F.&#125;}}
* ''infinity is approached from below'' if whenever {{mathimg|2f63bed964e89bbd2f00f55a26858febaa674872|width=6.508ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle F\in &#123;\mathcal &#123;F}} satisfies {{mathimg|19a2273e17a8e735abe20c50d054e6c2d37b067a|width=10.374ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (F)&#61;\infty &#125;}} then for every real {{mathimg|72ee1bdad381ab757caae14bc628dd1f9005fa5f|width=5.956ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle r>0,&#125;}} there exists some {{mathimg|aebed5ed3330f805b2480fff3a1b5e61b3479ee2|width=7.236ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;r&#125;\in &#123;\mathcal &#123;F}} such that {{mathimg|7def62f2e77fc40afbb5d0e3781612536a3fdcc3|width=7.308ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;r&#125;\subseteq F&#125;}} and {{mathimg|caa0b0155e65799bf294dd6041724e0f9b584c2e|width=16.283ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle r\leq \mu \left(F_&#123;r&#125;\right)&lt;\infty .&#125;}}
* an [[#outer measure|''outer measure'']] if {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is non-negative, [[#countably subadditive|countably subadditive]], has a [[#null empty set|null empty set]], and has the [[power set]] {{mathimg|d598c7f2d38b32ee7f7c4f0ad2fe9f4acfc5a170|width=4.966ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \wp (\Omega )&#125;}} as its domain.
* an [[Inner measure|''inner measure'']] if {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is non-negative, [[#superadditive|superadditive]], [[#continuous from above|continuous from above]], has a [[#null empty set|null empty set]], has the [[power set]] {{mathimg|d598c7f2d38b32ee7f7c4f0ad2fe9f4acfc5a170|width=4.966ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \wp (\Omega )&#125;}} as its domain, and [[#infinity is approached from below|{{mathimg|bddbb0e4420a7e744cf71bd71216e11b0bf88831|width=4.132ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle +\infty &#125;}} is approached from below]].
* [[Atomic measure|''atomic'']] if every measurable set of positive measure contains an [[Atom (measure theory)|atom]].

If a [[binary operation]] {{mathimg|31b3c23fb2245f1f8bc8dde4d8c42f582ce4d3e3|width=2.582ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle \,+\,&#125;}} is defined, then a set function {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is said to be

* ''[[translation invariant]]'' if {{mathimg|77c0472c05e72ffeb7c253987cd6d20e62bf8312|width=17.288ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (\omega +F)&#61;\mu (F)&#125;}} for all {{mathimg|e23a8931e1b954519d9fb9ba2e7f02eaa11ac91a|width=5.965ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \omega \in \Omega &#125;}} and {{mathimg|2f63bed964e89bbd2f00f55a26858febaa674872|width=6.508ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle F\in &#123;\mathcal &#123;F}} such that {{mathimg|ade00a894ff1c03e35ce9e4d481b64956cd9ad50|width=11.441ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle \omega +F\in &#123;\mathcal &#123;F.&#125;}}

=== Topology related definitions ===

If {{mathimg|38a7dcde9730ef0853809fefc18d88771f95206c|width=1.202ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \tau &#125;}} is a [[Topology (structure)|topology]] on {{mathimg|24b0d5ca6f381068d756f6337c08e0af9d1eeb6f|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \Omega &#125;}} then a set function {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is said to be:

* a [[Borel measure|''Borel measure'']] if it is a measure defined on the σ-algebra of all [[Borel set|Borel sets]], which is the smallest σ-algebra containing all open subsets (that is, containing {{mathimg|38a7dcde9730ef0853809fefc18d88771f95206c|width=1.202ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \tau &#125;}}).
* a [[Baire measure|''Baire measure'']] if it is a measure defined on the σ-algebra of all [[Baire set|Baire sets]].
* [[Locally finite measure|''locally finite'']] if for every point {{mathimg|e23a8931e1b954519d9fb9ba2e7f02eaa11ac91a|width=5.965ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \omega \in \Omega &#125;}} there exists some neighborhood {{mathimg|623423642d8e32bb2ec9642955937d11a0f68f65|width=10.334ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle U\in &#123;\mathcal &#123;F\cap \tau &#125;}} of this point such that {{mathimg|b98e66d47e84f346c0703076e6fdd0588bc3ff20|width=4.994ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (U)&#125;}} is finite.
** If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is a finitely additive, monotone, and locally finite then {{mathimg|2abfb9cecc1c69b28f9ba8c6fd743a01626ee3fe|width=5.277ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (K)&#125;}} is necessarily finite for every compact measurable subset {{mathimg|1bb0e178e42abf16ef4e4c0b0f22aa235ad6e6e5|width=2.713ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle K.&#125;}}
* [[Τ-additivity|''{{mathimg|38a7dcde9730ef0853809fefc18d88771f95206c|width=1.202ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \tau &#125;}}-additive'']] if {{mathimg|e7ee2ea32342dfb0811a1c58cbeea3f16d48c8b5|width=20.058ex|height=4.509ex|va=-2.505ex|alt=&#123;\displaystyle \mu \left(&#123;\textstyle \bigcup &#125;\,&#123;\mathcal &#123;D\right)&#61;\sup _&#123;D\in &#123;\mathcal &#123;D\mu (D)&#125;}} whenever {{mathimg|0d85586301b30da1981dab5de1e509798802f455|width=10.601ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle &#123;\mathcal &#123;D\subseteq \tau \cap &#123;\mathcal &#123;F}} is [[Directed set|directed]] with respect to {{mathimg|591976ad7ed25b287998b2c438d5391be58c5c18|width=2.582ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle \,\subseteq \,&#125;}} and satisfies {{mathimg|a9577898a7a113c34b5ed0b0f4dfe9e20f3ba46f|width=18.921ex|height=5.343ex|va=-2.671ex|alt=&#123;\displaystyle &#123;\textstyle \bigcup &#125;\,&#123;\mathcal &#123;D~&#123;\stackrel &#123;\scriptscriptstyle &#123;\text&#123;def&#123;&#61;~\textstyle \bigcup \limits _&#123;D\in &#123;\mathcal &#123;DD\in &#123;\mathcal &#123;F.&#125;}}
** {{mathimg|3277962e1959c3241fb1b70c7f0ac6dcefebd966|width=1.792ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;D}} is [[Directed set|directed]] with respect to {{mathimg|591976ad7ed25b287998b2c438d5391be58c5c18|width=2.582ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle \,\subseteq \,&#125;}} if and only if it is not empty and for all {{mathimg|a3a420004602d201ec89f9eddf240a12844cb142|width=9.174ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle A,B\in &#123;\mathcal &#123;D}} there exists some {{mathimg|4b59530b9e9e34b947980d6caaf5f194d0a5f8bc|width=6.399ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle C\in &#123;\mathcal &#123;D}} such that {{mathimg|c52a0a9fb646e916904c85763d980be597191ad2|width=6.608ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle A\subseteq C&#125;}} and {{mathimg|d17fc60cbf9fd462e651ffb004a24bd93c6aad5b|width=7.276ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle B\subseteq C.&#125;}}
* [[Inner regular measure|''inner regular'']] or ''tight'' if for every {{mathimg|b79bab9de399c6262d6d4197d8e48134b47d9c9c|width=7.155ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F\in &#123;\mathcal &#123;F,&#125;}} {{mathimg|7d7d7101b0e9cc4fcdd955988675521c86010840|width=67.7ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (F)&#61;\sup\&#123;\mu (K):F\supseteq K&#123;\text&#123; with K\in &#123;\mathcal &#123;F&#123;\text&#123; a compact subset of (\Omega ,\tau )\&#125;.&#125;}}
* [[Outer regular measure|''outer regular'']] if for every {{mathimg|b79bab9de399c6262d6d4197d8e48134b47d9c9c|width=7.155ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F\in &#123;\mathcal &#123;F,&#125;}} {{mathimg|389e776831d34f330cf448eff1e51bef9a8f5080|width=42.469ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (F)&#61;\inf\&#123;\mu (U):F\subseteq U&#123;\text&#123; and U\in &#123;\mathcal &#123;F\cap \tau \&#125;.&#125;}}
* [[Regular measure|''regular'']] if it is both inner regular and outer regular.
* a [[Borel regular measure|''Borel regular measure'']] if it is a Borel measure that is also [[Regular measure|''regular'']].
* a [[Radon measure|''Radon measure'']] if it is a regular and locally finite measure.
* [[Strictly positive measure|''strictly positive'']] if every non-empty open subset has (strictly) positive measure.
* a [[Valuation (measure theory)|''valuation'']] if it is non-negative, [[#monotone|monotone]], [[#modular|modular]], has a [[#null empty set|null empty set]], and has domain {{mathimg|871bb01391136d3551c8ea59059e106be2a403cd|width=1.849ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \tau .&#125;}}

=== Relationships between set functions ===

If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} and {{mathimg|c15bbbb971240cf328aba572178f091684585468|width=1.232ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \nu &#125;}} are two set functions over {{mathimg|b9e6a34b6c6b77b92a65f692aad0f65b10f5bf26|width=2.325ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle \Omega ,&#125;}} then:

* {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is said to be [[Absolute continuity (measure theory)|''absolutely continuous with respect to {{mathimg|c15bbbb971240cf328aba572178f091684585468|width=1.232ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \nu &#125;}}'']] or [[Domination (measure theory)|''dominated by {{mathimg|c15bbbb971240cf328aba572178f091684585468|width=1.232ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \nu &#125;}}'']], written {{mathimg|f26b16856bebfdea54c7dec9ec6b20dce7fd9d43|width=6.895ex|height=2.343ex|va=-0.838ex|alt=&#123;\displaystyle \mu \ll \nu ,&#125;}} if for every set {{mathimg|545fd099af8541605f7ee55f08225526be88ce57|width=1.741ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle F&#125;}} that belongs to the domain of both {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} and {{mathimg|a58da52de5fc23d5cc67b6a94171abb4420956f2|width=1.879ex|height=2.009ex|va=-0.671ex|alt=&#123;\displaystyle \nu ,&#125;}} if {{mathimg|79b18b2f27f4423cb8a28286ad48a1e1a457ad8b|width=9.043ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \nu (F)&#61;0&#125;}} then {{mathimg|83535ef1b0e557e968b577395fb154a14878d8e2|width=9.86ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (F)&#61;0.&#125;}}
** If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} and {{mathimg|c15bbbb971240cf328aba572178f091684585468|width=1.232ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \nu &#125;}} are [[Σ-finite measure|{{mathimg|59f59b7c3e6fdb1d0365a494b81fb9a696138c36|width=1.33ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \sigma &#125;}}-finite measures]] on the same measurable space and if {{mathimg|f26b16856bebfdea54c7dec9ec6b20dce7fd9d43|width=6.895ex|height=2.343ex|va=-0.838ex|alt=&#123;\displaystyle \mu \ll \nu ,&#125;}} then the [[Radon–Nikodym derivative]] {{mathimg|086efe6f61f7f451c5ff2821fe9deeb4ce6d720d|width=3.454ex|height=5.509ex|va=-2.005ex|alt=&#123;\displaystyle &#123;\frac &#123;d\mu &#125;&#123;d\nu }} exists and for every measurable {{mathimg|ed179a16e860b290a2c25c573d3fc0b2360f0fc1|width=2.388ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F,&#125;}} {{mathimg|12111d7e99285cd998ae5dbaaf4179666b580b75|width=17.742ex|height=5.843ex|va=-2.338ex|alt=&#123;\displaystyle \mu (F)&#61;\int _&#123;F&#125;&#123;\frac &#123;d\mu &#125;&#123;d\nu d\nu .&#125;}}
** {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} and {{mathimg|c15bbbb971240cf328aba572178f091684585468|width=1.232ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \nu &#125;}} are called [[Equivalence (measure theory)|''equivalent'']] if each one is [[#absolutely continuous|absolutely continuous]] with respect to the other. {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is called a [[Equivalence (measure theory)#Supporting measure|''supporting measure'']] of a measure {{mathimg|c15bbbb971240cf328aba572178f091684585468|width=1.232ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \nu &#125;}} if {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is [[Sigma-finite|{{mathimg|59f59b7c3e6fdb1d0365a494b81fb9a696138c36|width=1.33ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \sigma &#125;}}-finite]] and they are equivalent.<ref name="5">[[Olav Kallenberg|Kallenberg, Olav]] (2017). ''Random Measures, Theory and Applications''. Probability Theory and Stochastic Modelling. Vol.&#160;77. Switzerland: Springer. p.&#160;21. [[Doi (identifier)|doi]]:[https://doi.org/10.1007%2F978-3-319-41598-7 10.1007/978-3-319-41598-7]. [[ISBN (identifier)|ISBN]]&#160;<bdi>978-3-319-41596-3</bdi>.</ref>
* {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} and {{mathimg|c15bbbb971240cf328aba572178f091684585468|width=1.232ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \nu &#125;}} are [[Singular measure|''singular'']], written {{mathimg|4ed0907a62274e6db236c0a47acab06384c55760|width=6.379ex|height=2.676ex|va=-0.838ex|alt=&#123;\displaystyle \mu \perp \nu ,&#125;}} if there exist disjoint sets {{mathimg|f82cade9898ced02fdd08712e5f0c0151758a0dd|width=2.442ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle M&#125;}} and {{mathimg|f5e3890c981ae85503089652feb48b191b57aae3|width=2.064ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle N&#125;}} in the domains of {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} and {{mathimg|c15bbbb971240cf328aba572178f091684585468|width=1.232ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \nu &#125;}} such that {{mathimg|be5b85cf6af30bacf9a456b1c32252ec55ab3e29|width=12.512ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle M\cup N&#61;\Omega ,&#125;}} {{mathimg|dce9d9847db11c4acce7d478f4ef186f12fd9810|width=9.213ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (F)&#61;0&#125;}} for all {{mathimg|dd70b95bc89a924b3b519bf9a47d234d21c47981|width=7.281ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle F\subseteq M&#125;}} in the domain of {{mathimg|1e7e1ef161a49a22b500d63307460ad92eeb6a16|width=2.049ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu ,&#125;}} and {{mathimg|79b18b2f27f4423cb8a28286ad48a1e1a457ad8b|width=9.043ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \nu (F)&#61;0&#125;}} for all {{mathimg|e07fd7d5a48bb3056785883cb4afcd4f275fc2b0|width=6.903ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle F\subseteq N&#125;}} in the domain of {{mathimg|12bafc5c30bf9727eb4c005ff0c0632d555b3b7d|width=1.879ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \nu .&#125;}}

== Examples ==

Examples of set functions include:

* The function {{mathimg|c41cc484ae24573baf6f39b6f6c130ee6f57e164|width=29.689ex|height=5.676ex|va=-1.838ex|alt=&#123;\displaystyle d(A)&#61;\lim _&#123;n\to \infty &#125;&#123;\frac &#123;&#124;A\cap \&#123;1,\ldots ,n\&#125;&#124;&#125;&#123;n,&#125;}} assigning [[Natural density|densities]] to sufficiently [[well-behaved]] subsets {{mathimg|dfc9ff4aabd34a031c13279c5d2b314beb33af29|width=17.126ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle A\subseteq \&#123;1,2,3,\ldots \&#125;,&#125;}} is a set function.
* A [[probability measure]] assigns a probability to each set in a [[Sigma-algebra|σ-algebra]]. Specifically, the probability of the [[empty set]] is zero and the probability of the [[sample space]] is {{mathimg|9cc5fd8163a83100c5330622e9e317fa4e872403|width=1.809ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle 1,&#125;}} with other sets given probabilities between {{mathimg|2aae8864a3c1fec9585261791a809ddec1489950|width=1.162ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle 0&#125;}} and {{mathimg|af8c4e445819b13a052647aa3eb2be990b0a4b24|width=1.809ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle 1.&#125;}}
* 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]].

The [[Jordan measure]] on {{mathimg|c510b63578322050121fe966f2e5770bea43308d|width=2.897ex|height=2.343ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; ^&#123;n}} is a set function defined on the set of all Jordan measurable subsets of {{mathimg|a25a2fd68a4a1e81315789537b707856cfb09730|width=3.543ex|height=2.676ex|va=-0.671ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; ^&#123;n&#125;;&#125;}} it sends a Jordan measurable set to its Jordan measure.

=== Lebesgue measure ===

The [[Lebesgue measure]] on {{mathimg|786849c765da7a84dbc3cce43e96aad58a5868dc|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; &#125;}} is a set function that assigns a non-negative real number to every set of real numbers that belongs to the Lebesgue {{mathimg|59f59b7c3e6fdb1d0365a494b81fb9a696138c36|width=1.33ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \sigma &#125;}}-algebra.<ref name="6">Kolmogorov and Fomin 1975</ref>

Its definition begins with the set {{mathimg|ca14a76d239ba2a3f6e69d80f5d51a3fab3d049b|width=12.421ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \operatorname &#123;Intervals&#125; (\mathbb &#123;R&#125; )&#125;}} of all intervals of real numbers, which is a [[semialgebra]] on {{mathimg|dc9de9049e03e5e5a0cab57076dbe4a369c1e3a7|width=2.325ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; .&#125;}} The function that assigns to every interval {{mathimg|535ea7fc4134a31cbe2251d9d3511374bc41be9f|width=1.172ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle I&#125;}} its {{mathimg|1f76d03c661970ae83e4a5007bf2ac8900b011a2|width=9.312ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \operatorname &#123;length&#125; (I)&#125;}} is a finitely additive set function (explicitly, if {{mathimg|535ea7fc4134a31cbe2251d9d3511374bc41be9f|width=1.172ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle I&#125;}} has endpoints {{mathimg|41558abc50886fdf38817495b243958d7b3dd39b|width=5.326ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle a\leq b&#125;}} then {{mathimg|72ac416d9e8cb7520eec87656ad9902f06361561|width=17.479ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \operatorname &#123;length&#125; (I)&#61;b-a&#125;}}). This set function can be extended to the [[Lebesgue outer measure]] on {{mathimg|0522388d36b55de7babe4bbfc49475eaf590c2bd|width=2.325ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; ,&#125;}} which is the translation-invariant set function {{mathimg|ba032c4211f263003bd7ba980409adee19243c9a|width=17.966ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \lambda ^&#123;\!*\!&#125;:\wp (\mathbb &#123;R&#125; )\to &#91;0,\infty &#93;&#125;}} that sends a subset {{mathimg|651e39fa02a0a98bc7f719fb35a883abe09bd8c5|width=6.552ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle E\subseteq \mathbb &#123;R&#125; &#125;}} to the [[infimum]] {{mathimg|ea3e36c7149b73f44c3b18139541943a2db30d20|width=85.838ex|height=7.509ex|va=-3.171ex|alt=&#123;\displaystyle \lambda ^&#123;\!*\!&#125;(E)&#61;\inf \left\&#123;\sum _&#123;k&#61;1&#125;^&#123;\infty &#125;\operatorname &#123;length&#125; (I_&#123;k&#125;):&#123;(I_&#123;k&#125;)_&#123;k\in \mathbb &#123;N&#125; &#123;\text&#123; is a sequence of open intervals with E\subseteq \bigcup _&}} Lebesgue outer measure is not countably additive (and so is not a measure) although its restriction to the [[Sigma-algebra|𝜎-algebra]] of all subsets {{mathimg|a24d575fec8f3042eaa6ce20277976806f932448|width=7.219ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle M\subseteq \mathbb &#123;R&#125; &#125;}} that satisfy the [[Carathéodory's criterion|Carathéodory criterion]]: {{mathimg|50aac0f445e0c4371393a8cba183c192c60acaf3|width=51.836ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \lambda ^&#123;\!*\!&#125;(M)&#61;\lambda ^&#123;\!*\!&#125;(M\cap E)+\lambda ^&#123;\!*\!&#125;(M\cap E^&#123;c&#125;)\quad &#123;\text&#123; for every S\subseteq \mathbb &#123;R&#125; &#125;}} is a measure that called [[Lebesgue measure]]. [[Vitali set|Vitali sets]] are examples of [[Non-measurable set|non-measurable sets]] of real numbers.

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

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

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

The only translation-invariant measure on {{mathimg|fa509a134d45e796a6c29cf98237b4a2ff85fd8e|width=6.455ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \Omega &#61;\mathbb &#123;R&#125; &#125;}} with domain {{mathimg|6271848da40d4b8be111f20bd3381bdaeaaa925d|width=4.966ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \wp (\mathbb &#123;R&#125; )&#125;}} that is finite on every compact subset of {{mathimg|786849c765da7a84dbc3cce43e96aad58a5868dc|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; &#125;}} is the trivial set function {{mathimg|db085b81d6c70bb7d2cd71dca5f9e8ab09c36d31|width=14.394ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \wp (\mathbb &#123;R&#125; )\to &#91;0,\infty &#93;&#125;}} that is identically equal to {{mathimg|2aae8864a3c1fec9585261791a809ddec1489950|width=1.162ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle 0&#125;}} (that is, it sends every {{mathimg|2a8d8e251de9443239f3393459a3f0558a37507a|width=6.276ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle S\subseteq \mathbb &#123;R&#125; &#125;}} to {{mathimg|2aae8864a3c1fec9585261791a809ddec1489950|width=1.162ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle 0&#125;}})<ref name="FOOTNOTERudin1991139-7">[[#CITEREFRudin1991|Rudin 1991]], p.&#160;139.</ref> 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 {{mathimg|8786b5ef9daedb24adb59e7825c4096d99a99648|width=5.299ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#91;0,1&#93;.&#125;}} In fact, such non-trivial set functions will exist even if {{mathimg|786849c765da7a84dbc3cce43e96aad58a5868dc|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; &#125;}} is replaced by any other [[Abelian group|abelian]] [[Group (mathematics)|group]] {{mathimg|dc645a5b7e8a2022ad70fc42dbda04c008a33a9a|width=2.474ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle G.&#125;}}<ref name="FOOTNOTERudin1991139–140-8">[[#CITEREFRudin1991|Rudin 1991]], pp.&#160;139–140.</ref>

'''Theorem<ref name="FOOTNOTERudin1991141–142-10">[[#CITEREFRudin1991|Rudin 1991]], pp.&#160;141–142.</ref>'''—If {{mathimg|ede3320938add8f227928a5efe5738feaa903345|width=6.478ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle (G,+)&#125;}} is any [[abelian group]] then there exists a finitely additive and translation-invariant<ref name="GroupTranslation-9">The function {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} being translation-invariant means that {{mathimg|ad46e02e6434d9f56dc64702cd6a5314d401bd37|width=16.475ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (S)&#61;\mu (g+S)&#125;}} for every {{mathimg|b1be73903416a0dd94b8cbc2268ce480810c0e62|width=5.783ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle g\in G&#125;}} and every subset {{mathimg|778fd7e9a21ff21c5c3f08c1f252b69ccbf66c66|width=7.071ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle S\subseteq G.&#125;}}</ref> set function {{mathimg|c467807cd159395f4021fc800100db9aaf263fdd|width=16.72ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu :\wp (G)\to &#91;0,1&#93;&#125;}} of mass {{mathimg|24a47a59496a4ebbd5735ff0990b453ec745ccc2|width=9.945ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu (G)&#61;1.&#125;}}

== Extending set functions ==

=== Extending from semialgebras to algebras ===

Suppose that {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is a set function on a [[semialgebra]] {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} over {{mathimg|24b0d5ca6f381068d756f6337c08e0af9d1eeb6f|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \Omega &#125;}} and let {{mathimg|db2f3ccf726385e912689dbf3aa73434e6b54494|width=80.618ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \operatorname &#123;algebra&#125; (&#123;\mathcal &#123;F):&#61;\left\&#123;F_&#123;1&#125;\sqcup \cdots \sqcup F_&#123;n&#125;:n\in \mathbb &#123;N&#125; &#123;\text&#123; and F_&#123;1&#125;,\ldots ,F_&#123;n&#125;\in &#123;\mathcal &#123;F&#123;\text&#123; are pairwise disjoin}} which is the [[Field of sets|algebra]] on {{mathimg|24b0d5ca6f381068d756f6337c08e0af9d1eeb6f|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \Omega &#125;}} generated by {{mathimg|6e1656ae73ede684468b360e948a8a38e6e2c461|width=2.573ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F.&#125;}} The archetypal example of a semialgebra that is not also an [[Field of sets|algebra]] is the family {{mathimg|0b6ed20f1389f1fd2a8773cdae4daec438d2aaf1|width=80.07ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#123;\mathcal &#123;S_&#123;d&#125;:&#61;\&#123;\varnothing \&#125;\cup \left\&#123;\left(a_&#123;1&#125;,b_&#123;1&#125;\right&#93;\times \cdots \times \left(a_&#123;1&#125;,b_&#123;1&#125;\right&#93;~:~-\infty \leq a_&#123;i&#125;&lt;b_&#123;i&#125;\leq \infty &#123;\text&#123}} on {{mathimg|3101234d4318238624a292393d5327801513bdda|width=8.193ex|height=2.676ex|va=-0.338ex|alt=&#123;\displaystyle \Omega :&#61;\mathbb &#123;R&#125; ^&#123;d}} where {{mathimg|01b917151b48c11cecabffef134fb5bb6f07c939|width=28.423ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle (a,b&#93;:&#61;\&#123;x\in \mathbb &#123;R&#125; :a&lt;x\leq b\}} for all {{mathimg|82fff6b8e90b71ad40b732073b0385b3c445aceb|width=18.625ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle -\infty \leq a&lt;b\leq \infty .&#125;}}<ref name="FOOTNOTEDurrett20191–9-11">[[#CITEREFDurrett2019|Durrett 2019]], pp.&#160;1–9.</ref> Importantly, the two non-strict inequalities {{mathimg|24112548985eab096493f73f838580442780b57f|width=2.582ex|height=2.176ex|va=-0.505ex|alt=&#123;\displaystyle \,\leq \,&#125;}} in {{mathimg|1df9c5e7eecd32ed5278186bd863553c36b49de7|width=19.578ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle -\infty \leq a_&#123;i&#125;&lt;b_&#123;i&#125;\leq \infty &#125;}} cannot be replaced with strict inequalities {{mathimg|f5ebb5b330e53c9b9af8e7d7c8e0590d3a5f631e|width=2.582ex|height=1.843ex|va=-0.338ex|alt=&#123;\displaystyle \,&lt;\,&#125;}} since semialgebras must contain the whole underlying set {{mathimg|b8f67a6b6643c72491aee98a1ab84daa229cdf96|width=3.417ex|height=3.009ex|va=-0.671ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; ^&#123;d&#125;;&#125;}} that is, {{mathimg|aa16c109b0a6c0e8054812e37c7efe34e18df565|width=8.111ex|height=3.009ex|va=-0.671ex|alt=&#123;\displaystyle \mathbb &#123;R&#125; ^&#123;d&#125;\in &#123;\mathcal &#123;S_&#123;d}} is a requirement of semialgebras (as is {{mathimg|f0e44af9e5e605d2e659a14a57dc43b10203a7e5|width=7.149ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle \varnothing \in &#123;\mathcal &#123;S_&#123;d}}).

If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is [[#finitely additive|finitely additive]] then it has a unique extension to a set function {{mathimg|7b4b3a49dc0539aaf0e10b3793cb06ad503e17e1|width=1.517ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#123;\overline &#123;\mu }} on {{mathimg|0f2208bb34eb75ae715ee50ed72ea5bf4541c4d5|width=11.107ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \operatorname &#123;algebra&#125; (&#123;\mathcal &#123;F)&#125;}} defined by sending {{mathimg|995cdda50e53c00e49f912a45b7a8d8c58c9c085|width=27.098ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle F_&#123;1&#125;\sqcup \cdots \sqcup F_&#123;n&#125;\in \operatorname &#123;algebra&#125; (&#123;\mathcal &#123;F)&#125;}} (where {{mathimg|8fcd225879fcce9a8f7b356f373f5c0732a9ab2e|width=2.325ex|height=2.009ex|va=-0.338ex|alt=&#123;\displaystyle \,\sqcup \,&#125;}} indicates that these {{mathimg|6d910243a53b386e59647997b555770b5a77a9f0|width=7.062ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle F_&#123;i&#125;\in &#123;\mathcal &#123;F}} are [[pairwise disjoint]]) to:<ref name="FOOTNOTEDurrett20191–9-11"/> {{mathimg|dcf403189b521a472dd30022672452e5a51be12a|width=42.505ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#123;\overline &#123;\mu \left(F_&#123;1&#125;\sqcup \cdots \sqcup F_&#123;n&#125;\right):&#61;\mu \left(F_&#123;1&#125;\right)+\cdots +\mu \left(F_&#123;n&#125;\right).&#125;}} This extension {{mathimg|7b4b3a49dc0539aaf0e10b3793cb06ad503e17e1|width=1.517ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#123;\overline &#123;\mu }} will also be finitely additive: for any pairwise disjoint {{mathimg|fbfe287e5370c1d35841ea4888dceb95a32cc484|width=25.531ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle A_&#123;1&#125;,\ldots ,A_&#123;n&#125;\in \operatorname &#123;algebra&#125; (&#123;\mathcal &#123;F),&#125;}} <ref name="FOOTNOTEDurrett20191–9-11"/> {{mathimg|6e3547b4cbf8c74abd9570efab36f375369b4481|width=43.082ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#123;\overline &#123;\mu \left(A_&#123;1&#125;\cup \cdots \cup A_&#123;n&#125;\right)&#61;&#123;\overline &#123;\mu \left(A_&#123;1&#125;\right)+\cdots +&#123;\overline &#123;\mu \left(A_&#123;n&#125;\right).&#125;}}

If in addition {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is extended real-valued and [[#monotone|monotone]] (which, in particular, will be the case if {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is [[#non-negative|non-negative]]) then {{mathimg|7b4b3a49dc0539aaf0e10b3793cb06ad503e17e1|width=1.517ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#123;\overline &#123;\mu }} will be monotone and [[#finitely subadditive|finitely subadditive]]: for any {{mathimg|c2f5de299ec174b71920a90e2279cff910fc28e2|width=27.661ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle A,A_&#123;1&#125;,\ldots ,A_&#123;n&#125;\in \operatorname &#123;algebra&#125; (&#123;\mathcal &#123;F)&#125;}} such that {{mathimg|43cb0fad8699b1a70e7bd1b820334e890ca858b0|width=19.136ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle A\subseteq A_&#123;1&#125;\cup \cdots \cup A_&#123;n&#125;,&#125;}}<ref name="FOOTNOTEDurrett20191–9-11"/> {{mathimg|d6ac331890e85bf668dee544ca4bfa320de146dc|width=31.178ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#123;\overline &#123;\mu \left(A\right)\leq &#123;\overline &#123;\mu \left(A_&#123;1&#125;\right)+\cdots +&#123;\overline &#123;\mu \left(A_&#123;n&#125;\right).&#125;}}

=== Extending from rings to σ-algebras ===

If {{mathimg|3d70b2697d710964ae838c95e96295f704b5d3c3|width=14.693ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu :&#123;\mathcal &#123;F\to &#91;0,\infty &#93;&#125;}} is a [[#pre-measure|pre-measure]] on a [[ring of sets]] (such as an [[algebra of sets]]) {{mathimg|205d4b91000d9dcf1a5bbabdfa6a8395fa60b676|width=1.927ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F}} over {{mathimg|24b0d5ca6f381068d756f6337c08e0af9d1eeb6f|width=1.678ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle \Omega &#125;}} then {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} has an extension to a measure {{mathimg|65d2355f01636d067f52141b7999ef767170d568|width=17.947ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle &#123;\overline &#123;\mu :\sigma (&#123;\mathcal &#123;F)\to &#91;0,\infty &#93;&#125;}} on the [[σ-algebra]] {{mathimg|0cda4b539eb8abd6d71c580bf2247f9a0f933721|width=5.066ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \sigma (&#123;\mathcal &#123;F)&#125;}} generated by {{mathimg|6e1656ae73ede684468b360e948a8a38e6e2c461|width=2.573ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F.&#125;}} If {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} is [[#σ-finite measure|σ-finite]] then this extension is unique.

To define this extension, first extend {{mathimg|9fd47b2a39f7a7856952afec1f1db72c67af6161|width=1.402ex|height=2.176ex|va=-0.838ex|alt=&#123;\displaystyle \mu &#125;}} to an [[outer measure]] {{mathimg|670d0d4db6668c13d249c92fb99c34d2a9f236f7|width=2.456ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu ^&#123;*}} on {{mathimg|5ae55706515676005bfff286251a16ad052dc671|width=10.645ex|height=3.176ex|va=-0.838ex|alt=&#123;\displaystyle 2^&#123;\Omega &#125;&#61;\wp (\Omega )&#125;}} by {{mathimg|866387b3c3bc1201c24e1ccf4f168d8dcef4b841|width=58.052ex|height=7.509ex|va=-3.171ex|alt=&#123;\displaystyle \mu ^&#123;*&#125;(T)&#61;\inf \left\&#123;\sum _&#123;n&#125;\mu \left(S_&#123;n&#125;\right):T\subseteq \cup _&#123;n&#125;S_&#123;n&#125;&#123;\text&#123; with S_&#123;1&#125;,S_&#123;2&#125;,\ldots \in &#123;\mathcal &#123;F\right\}} and then restrict it to the set {{mathimg|22e4eeb9eee27d7ec0b555e6135f7bbd0a0db2ae|width=3.63ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle &#123;\mathcal &#123;F_&#123;M}} of {{mathimg|670d0d4db6668c13d249c92fb99c34d2a9f236f7|width=2.456ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu ^&#123;*}}-measurable sets (that is, [[Carathéodory-measurable set|Carathéodory-measurable sets]]), which is the set of all {{mathimg|bdc2fb48070208807e2b009ce7bdd7b7e204b10c|width=7.219ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle M\subseteq \Omega &#125;}} such that {{mathimg|ba8a5b724c63aef23abe6c4ba0a0e5502a50badb|width=60.44ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu ^&#123;*&#125;(S)&#61;\mu ^&#123;*&#125;(S\cap M)+\mu ^&#123;*&#125;(S\cap M^&#123;\mathrm &#123;c&#125; &#125;)\quad &#123;\text&#123; for every subset S\subseteq \Omega .&#125;}} It is a {{mathimg|59f59b7c3e6fdb1d0365a494b81fb9a696138c36|width=1.33ex|height=1.676ex|va=-0.338ex|alt=&#123;\displaystyle \sigma &#125;}}-algebra and {{mathimg|670d0d4db6668c13d249c92fb99c34d2a9f236f7|width=2.456ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu ^&#123;*}} is sigma-additive on it, by Caratheodory lemma.

=== Restricting outer measures ===

If {{mathimg|ec617e33964898d457a5be8517ac70004caf2a2d|width=18.787ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu ^&#123;*&#125;:\wp (\Omega )\to &#91;0,\infty &#93;&#125;}} is an [[#outer measure|outer measure]] on a set {{mathimg|b9e6a34b6c6b77b92a65f692aad0f65b10f5bf26|width=2.325ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle \Omega ,&#125;}} where (by definition) the domain is necessarily the [[power set]] {{mathimg|d598c7f2d38b32ee7f7c4f0ad2fe9f4acfc5a170|width=4.966ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \wp (\Omega )&#125;}} of {{mathimg|b9e6a34b6c6b77b92a65f692aad0f65b10f5bf26|width=2.325ex|height=2.509ex|va=-0.671ex|alt=&#123;\displaystyle \Omega ,&#125;}} then a subset {{mathimg|bdc2fb48070208807e2b009ce7bdd7b7e204b10c|width=7.219ex|height=2.343ex|va=-0.505ex|alt=&#123;\displaystyle M\subseteq \Omega &#125;}} is called '''''{{mathimg|670d0d4db6668c13d249c92fb99c34d2a9f236f7|width=2.456ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu ^&#123;*}}–measurable''''' or '''''[[Carathéodory-measurable set|Carathéodory-measurable]]''''' if it satisfies the following ''[[Carathéodory's criterion]]'': {{mathimg|841892639e2ca974937797d93f9f21208d76d67c|width=60.44ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu ^&#123;*&#125;(S)&#61;\mu ^&#123;*&#125;(S\cap M)+\mu ^&#123;*&#125;(S\cap M^&#123;\mathrm &#123;c&#125; &#125;)\quad &#123;\text&#123; for every subset S\subseteq \Omega ,&#125;}} where {{mathimg|2866017276000f5425b36e9efa3c68c6da07ef01|width=13.521ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle M^&#123;\mathrm &#123;c&#125; &#125;:&#61;\Omega \setminus M&#125;}} is the [[Complement (set theory)|complement]] of {{mathimg|b865c33e30eb83000cd6387517c66dbbf3c3df9e|width=3.089ex|height=2.176ex|va=-0.338ex|alt=&#123;\displaystyle M.&#125;}}

The family of all {{mathimg|670d0d4db6668c13d249c92fb99c34d2a9f236f7|width=2.456ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu ^&#123;*}}–measurable subsets is a [[σ-algebra]] and the [[Restriction (mathematics)|restriction]] of the outer measure {{mathimg|670d0d4db6668c13d249c92fb99c34d2a9f236f7|width=2.456ex|height=2.843ex|va=-0.838ex|alt=&#123;\displaystyle \mu ^&#123;*}} to this family is a [[Measure (mathematics)|measure]].

== See also ==

* [[Absolute continuity (measure theory)]]&#160;– Form of continuity for functions
* [[Boolean ring]]&#160;– Algebraic structure in mathematics
* [[Cylinder set measure]]
* [[Field of sets]]&#160;– Algebraic concept in measure theory, also referred to as an algebra of sets
* [[Hadwiger's theorem]]&#160;– Theorem in integral geometry
* [[Hahn decomposition theorem]]&#160;– Measurability theorem
* [[Invariant measure]]&#160;– Concept in mathematics
* [[Lebesgue's decomposition theorem]]&#160;– Theorem in mathematical measure theory
* [[Positive and negative sets]]&#160;– Concept in measure theory
* [[Radon–Nikodym theorem]]&#160;– Expressing a measure as an integral of another
* [[Riesz–Markov–Kakutani representation theorem]]&#160;– Statement about linear functionals and measures
* [[Ring of sets]]&#160;– Family closed under unions and relative complements
* [[σ-algebra]]&#160;– Algebraic structure of set algebra
* [[Vitali–Hahn–Saks theorem]]

== Notes ==

'''Proofs'''

== References ==

* [[Richard Durrett|Durrett, Richard]] (2019). [https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf ''Probability: Theory and Examples''] (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol.&#160;49 (5th&#160;ed.). Cambridge New York, NY: [[Cambridge University Press]]. [[ISBN (identifier)|ISBN]]&#160;<bdi>978-1-108-47368-2</bdi>. [[OCLC (identifier)|OCLC]]&#160;[https://search.worldcat.org/oclc/1100115281 1100115281]. Retrieved November 5, 2020.
* [[Andrey Kolmogorov|Kolmogorov, Andrey]]; [[Sergei Fomin|Fomin, Sergei V.]] (2012) &#91;1957&#93;. ''Elements of the Theory of Functions and Functional Analysis''. Dover Books on Mathematics. New York: Dover Books. [[ISBN (identifier)|ISBN]]&#160;<bdi>978-1-61427-304-2</bdi>. [[OCLC (identifier)|OCLC]]&#160;[https://search.worldcat.org/oclc/912495626 912495626].
* A. N. Kolmogorov and S. V. Fomin (1975), ''Introductory Real Analysis'', Dover. [[ISBN (identifier)|ISBN]]&#160;<bdi>0-486-61226-0</bdi>
* [[Halsey Royden|Royden, Halsey]]; [[Patrick Fitzpatrick|Fitzpatrick, Patrick]] (15 January 2010). ''Real Analysis'' (4&#160;ed.). Boston: [[Prentice Hall]]. [[ISBN (identifier)|ISBN]]&#160;<bdi>978-0-13-143747-0</bdi>. [[OCLC (identifier)|OCLC]]&#160;[https://search.worldcat.org/oclc/456836719 456836719].
* [[Walter Rudin|Rudin, Walter]] (1991). [https://archive.org/details/functionalanalys00rudi ''Functional Analysis'']. International Series in Pure and Applied Mathematics. Vol.&#160;8 (Second&#160;ed.). New York, NY: [[McGraw-Hill Science/Engineering/Math]]. [[ISBN (identifier)|ISBN]]&#160;<bdi>978-0-07-054236-5</bdi>. [[OCLC (identifier)|OCLC]]&#160;[https://search.worldcat.org/oclc/21163277 21163277].

== Further reading ==

* Sobolev, V.I. (2001) &#91;1994&#93;, [https://www.encyclopediaofmath.org/index.php?title=Set_function "Set function"], ''[[Encyclopedia of Mathematics]]'', EMS Press
* [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)]]

Preview

The preview appears here as you type.

Edit summary (briefly describe your change)

Cancel

By publishing, you agree to release your contribution under the CC BY-SA 4.0 License.