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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s0913001.png" />''
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{{MSC|28C15}}
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s0913002.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s0913003.png" /> is a locally compact Hausdorff space on which the regular Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s0913004.png" /> is given and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s0913005.png" /> is the largest open set for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s0913006.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s0913007.png" /> is the smallest closed set on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s0913008.png" /> is concentrated. (Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s0913009.png" /> is concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130010.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130011.png" />.) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130012.png" /> is compact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130013.png" /> is called of compact support.
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{{TEX|done}}
  
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[[Category:Set functions and measures on topological spaces]]
  
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Let $X$ be a topological space with a countable [[Basis|basis]], $\mathcal{B}$ a $\sigma$-algebra of subsets of $X$ containing the open sets (and hence also the [[Borel set|Borel sets]]) and $\mu: \mathcal{B}\to [0, \infty]$ a ($\sigma$-additive) measure (cp. with [[Borel measure]]). The support of $\mu$ (usually denoted by ${\rm supp}\, (\mu)$) is the complement of the union of all open sets which are $\mu$-null sets, i.e. the smallest closed set $C$ such that $\mu (X\setminus C) =0$.
  
====Comments====
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The existence of a countable base guarantees that
The support of a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130014.png" /> on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130015.png" /> can be defined whenever the union of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130016.png" />-zero open subsets is still of measure zero. This is the case if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130017.png" /> has a countable base, or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130018.png" /> is tight or Radon (see [[Regular measure|Regular measure]]), but it is not always the case if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130019.png" /> is only locally compact and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130020.png" /> is not tight.
 
  
Of course, one can always define, for a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130021.png" /> on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130022.png" /> with topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130023.png" />,
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(P) the union of all open $\mu$-null sets is itself a nullset.
  
<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/s091/s091300/s09130024.png" /></td> </tr></table>
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If (P) does not hold and $C$ is the complement of the union of all open sets which are $\mu$-null sets, then we do not have $\mu (X\setminus C)=0$.
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When $X$ does not have a countable base, property (P) can still be inferred from other assumptions. A common one is to assume that $X$ is a locally compact Hausdorff space and $\mu$ is a [[Radon measure]] (some authors use for such measures the terminologies tight and Borel regular, compare with [[Borel measure]]). For such
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measures the support is then defined as above.
  
But then it is not necessarily true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091300/s09130025.png" />, contradicting the intuitive idea of a support.
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The support can be defined also in case $\mu$ is a [[Signed measure|signed measure]] or a [[Vector measure|vector measure]] (in both cases we are assuming $\sigma$-additivity): the support of $\mu$ is then defined as the support of its total variation measure (see [[Signed measure]] for the definition).
  
====References====
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===References===
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1966) pp. 57</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Federer,  "Geometric measure theory" , Springer (1969pp. 60; 62; 71; 108</TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|AFP}}||  L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations  and  free discontinuity problems". Oxford Mathematical Monographs. The  Clarendon Press, Oxford University Press, New York, 2000.  {{MR|1857292}}{{ZBL|0957.49001}}
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|-
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|valign="top"|{{Ref|Bi}}||  P. Billingsley, "Convergence of  probability measures" , Wiley (1968)  {{MR|0233396}} {{ZBL|0172.21201}}
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|-
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|valign="top"|{{Ref|Bo}}||  N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley  (1975) pp. Chapt.6;7;8 (Translated from French) {{MR|0583191}}  {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}}  {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}}  {{ZBL|1095.28001}} {{ZBL|0156.06001}}
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|-
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|valign="top"|{{Ref|DS}}||  N. Dunford, J.T. Schwartz, "Linear operators. General theory" ,  '''1'''  , Interscience (1958) {{MR|0117523}}
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|-
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|valign="top"|{{Ref|Ha}}|| P.R. Halmos,  "Measure theory", v. Nostrand (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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|-
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|valign="top"|{{Ref|Ma}}||  P. Mattila, "Geometry of sets and measures in euclidean spaces.  Cambridge Studies in Advanced Mathematics, 44. Cambridge University  Press, Cambridge, 1995. {{MR|1333890}} {{ZBL|0911.28005}}
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|-
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|}

Latest revision as of 16:15, 29 November 2012

2020 Mathematics Subject Classification: Primary: 28C15 [MSN][ZBL]

Let $X$ be a topological space with a countable basis, $\mathcal{B}$ a $\sigma$-algebra of subsets of $X$ containing the open sets (and hence also the Borel sets) and $\mu: \mathcal{B}\to [0, \infty]$ a ($\sigma$-additive) measure (cp. with Borel measure). The support of $\mu$ (usually denoted by ${\rm supp}\, (\mu)$) is the complement of the union of all open sets which are $\mu$-null sets, i.e. the smallest closed set $C$ such that $\mu (X\setminus C) =0$.

The existence of a countable base guarantees that

(P) the union of all open $\mu$-null sets is itself a nullset.

If (P) does not hold and $C$ is the complement of the union of all open sets which are $\mu$-null sets, then we do not have $\mu (X\setminus C)=0$. When $X$ does not have a countable base, property (P) can still be inferred from other assumptions. A common one is to assume that $X$ is a locally compact Hausdorff space and $\mu$ is a Radon measure (some authors use for such measures the terminologies tight and Borel regular, compare with Borel measure). For such measures the support is then defined as above.

The support can be defined also in case $\mu$ is a signed measure or a vector measure (in both cases we are assuming $\sigma$-additivity): the support of $\mu$ is then defined as the support of its total variation measure (see Signed measure for the definition).

References

[AFP] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Bi] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201
[Bo] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001
[DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523
[Ha] P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Ma] P. Mattila, "Geometry of sets and measures in euclidean spaces. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005
How to Cite This Entry:
Support of a measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Support_of_a_measure&oldid=12419
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article