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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s0847301.png" /> of a certain collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s0847302.png" /> of subsets of a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s0847303.png" /> into another set, usually into the real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s0847304.png" /> or the complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s0847305.png" />. An important class of set functions are the additive set functions, for which
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s0847306.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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{{MSC|28A}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s0847307.png" /></td> </tr></table>
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[[Category:Classical measure theory]]
  
and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s0847309.png" />-additive set functions, which satisfy equation (*) for a countably infinite collection of sets also (replace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s08473010.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s08473011.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s08473012.png" /> takes only non-negative values, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s08473013.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s08473014.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s08473015.png" />-algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084730/s08473016.png" /> is called a [[Measure|measure]].
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{{TEX|done}}
  
====References====
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A mapping $\mu$ defined on a family $\mathcal{S}$ of subsets of a set $X$. Commonly the target of $\mu$ is a topological vector space $V$ (more generally a commutative topological group) or the extended real line $[-\infty, \infty]$ (in the latter case, to avoid operations of type $\infty + (-\infty)$ it is assumed that the range is either contained in $[-\infty, \infty[$ or in $]-\infty, \infty]$). It is usually assumed that the empty set is an element of $\mathcal{S}$ and that $\mu (\emptyset) =0$.
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.V. Kantorovich,   G.P. Akilov,   "Functional analysis" , Pergamon  (1982) (Translated from Russian)</TD></TR></table>
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 +
Notable examples are
  
 +
* Finitely additive set functions. In this case the domain of definition is a [[Ring of sets|ring]] (more often an [[Algebra of sets|algebra]]) and $\mu$ has the property that
 +
\[
 +
\mu \left(\bigcup_{i=1}^N E_i\right) = \sum_{i=1}^N \mu (E_i)
 +
\]
 +
for every finite collection $\{E_i\}$ of ''disjoint'' elements of $\mathcal{S}$.
  
 +
* Measures. In this case the domain of definition $\mathcal{S}$ is a [[Ring of sets|$\sigma$-ring]] (more often a [[Algebra of sets|$\sigma$-algebra]]) and the set function is assumed to be ''$\sigma$-additive'' (or, equivalently ''countably additive''), that is
 +
\[
 +
\mu \left(\bigcup_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \mu (E_i)
 +
\]
 +
for every countable collection $\{E_i\}$ of ''disjoint'' elements of $\mathcal{S}$. Note that, since we assume $\mu (\emptyset) = 0$, a measure is always finitely additive.
  
====Comments====
+
The word [[Measure|measure]] is indeed commonly used for such set functions which are taking values in $[0, \infty]$ and if in addition $\mu (X)=1$, then $\mu$ is a [[Probability measure|probability measure]]. $\sigma$-additive set functions taking values in the extended real line $[-\infty, \infty]$ are commonly called [[Signed measure|signed measures]] (some authors use also the name ''charge''), whereas $\sigma$-additive set functions taking values in vector spaces are commonly called [[Vector measure|vector measures]].
  
 +
*Outer measures. The domain of definition $\mathcal{S}$ of an [[Outer measure|outer measure]] $\mu$ is an hereditary $\sigma$-ring (also called $\sigma$-ideal), i.e. a $\sigma$-ring $\mathcal{S}$ with the additional property that it contains any subset of any of its elements (however, the most commonly used outer measures are defined on the whole space $\mathcal{P} (X)$ of all subsets of $X$). An outer measure takes values in $[0, \infty]$ and it is required to be ''$\sigma$-subadditive'' (or countably subadditive''), i.e.
 +
\[
 +
\mu \left(\bigcup_{i=1}^\infty E_i\right) \leq \sum_{i=1}^\infty \mu (E_i)
 +
\]
 +
for every countable collection $\{E_i\}$ of subsets of $X$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience (1958)</TD></TR></table>
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{|
 +
|-
 +
|valign="top"|{{Ref|Ha}}|| P.R. Halmos, "Measure theory" , v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
 +
|-
 +
|valign="top"|{{Ref|DS}}|| N. Dunford, J.T. Schwartz, "Linear operators. General theory" , '''1''' , Interscience (1958) {{MR|0117523}}
 +
|-
 +
|}

Latest revision as of 14:45, 19 September 2012

2010 Mathematics Subject Classification: Primary: 28A [MSN][ZBL]

A mapping $\mu$ defined on a family $\mathcal{S}$ of subsets of a set $X$. Commonly the target of $\mu$ is a topological vector space $V$ (more generally a commutative topological group) or the extended real line $[-\infty, \infty]$ (in the latter case, to avoid operations of type $\infty + (-\infty)$ it is assumed that the range is either contained in $[-\infty, \infty[$ or in $]-\infty, \infty]$). It is usually assumed that the empty set is an element of $\mathcal{S}$ and that $\mu (\emptyset) =0$.

Notable examples are

  • Finitely additive set functions. In this case the domain of definition is a ring (more often an algebra) and $\mu$ has the property that

\[ \mu \left(\bigcup_{i=1}^N E_i\right) = \sum_{i=1}^N \mu (E_i) \] for every finite collection $\{E_i\}$ of disjoint elements of $\mathcal{S}$.

  • Measures. In this case the domain of definition $\mathcal{S}$ is a $\sigma$-ring (more often a $\sigma$-algebra) and the set function is assumed to be $\sigma$-additive (or, equivalently countably additive), that is

\[ \mu \left(\bigcup_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \mu (E_i) \] for every countable collection $\{E_i\}$ of disjoint elements of $\mathcal{S}$. Note that, since we assume $\mu (\emptyset) = 0$, a measure is always finitely additive.

The word measure is indeed commonly used for such set functions which are taking values in $[0, \infty]$ and if in addition $\mu (X)=1$, then $\mu$ is a probability measure. $\sigma$-additive set functions taking values in the extended real line $[-\infty, \infty]$ are commonly called signed measures (some authors use also the name charge), whereas $\sigma$-additive set functions taking values in vector spaces are commonly called vector measures.

  • Outer measures. The domain of definition $\mathcal{S}$ of an outer measure $\mu$ is an hereditary $\sigma$-ring (also called $\sigma$-ideal), i.e. a $\sigma$-ring $\mathcal{S}$ with the additional property that it contains any subset of any of its elements (however, the most commonly used outer measures are defined on the whole space $\mathcal{P} (X)$ of all subsets of $X$). An outer measure takes values in $[0, \infty]$ and it is required to be $\sigma$-subadditive (or countably subadditive), i.e.

\[ \mu \left(\bigcup_{i=1}^\infty E_i\right) \leq \sum_{i=1}^\infty \mu (E_i) \] for every countable collection $\{E_i\}$ of subsets of $X$.

References

[Ha] P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
How to Cite This Entry:
Set function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Set_function&oldid=15370
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article