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m (moved Mutually-singular measures to Singular measures: More common name. I will redirect the old page.)
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Two (positive) measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m0656101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m0656102.png" />, defined on a locally compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m0656103.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m0656104.png" />.
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{{MSC|28A33}}
 
 
Two measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m0656105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m0656106.png" /> are mutually singular if and only if there exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m0656107.png" /> two disjoint sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m0656108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m0656109.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m06561010.png" /> is concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m06561011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m06561012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m06561013.png" />.
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Integration" , Addison-Wesley  (1975)  pp. Chapt.6;7;8  (Translated from French)</TD></TR></table>
 
  
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[[Category:Classical measure theory]]
  
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{{TEX|done}}
  
====Comments====
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If $\mu$ and $\nu$ are two $\sigma$-finite measures on the same [[Algebra of sets|$\sigma$-algebra]] $\mathcal{B}$ of subsets of $X$, then $\mu$ and $\nu$ are said to be singular
The second characterization in the main article above holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m06561014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m06561015.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m06561016.png" />-additive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m06561017.png" />-finite measures on an abstract [[Measurable space|measurable space]], and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m06561018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m06561019.png" /> belong to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065610/m06561020.png" />-field.
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(or also mutually singular, or orthogonal) if there are two sets $A,B\in\mathcal{B}$ such that $A\cap B=\emptyset$, $A\cup B = X$ and $\mu (B)=\nu (A) = 0$. The concept can be extended to [[Signed measure|signed measures]] or vector-valued measures: in this case it is required that $\mu (B\cap E) = \nu (A\cap E) = 0$ for every $E\in\mathcal{B}$ (cp. with Section 30 of {{Cite|Ha}}). The singularity of the two measures $\mu$ and $\nu$ is usually denoted by $\mu\perp\nu$.
  
Mutually-singular measures are also called singular measures or orthogonal measures.
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For general, i.e. non $\sigma$-finite (nonnegative) measures, the concept can be generalized in the following way: $\mu$ and $\nu$ are singular if the only (nonnegative) measure $\alpha$ on $\mathcal{B}$ with the property
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\[
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\alpha (A)\leq \min \{\mu (A), \nu (A)\} \qquad \forall A\in\mathcal{B}
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\]
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is the trivial measure which assigns the value $0$ to every element of $\mathcal{B}$. By the [[Absolutely continuous measures|Radon-Nikodym decomposition]] this concept coincides with the previous one when we assume the $\sigma$-finiteness of $\mu$ and $\nu$.
  
Instead of  "concentrated on"  one also uses  "supported in"  (cf. also [[Support of a measure|Support of a measure]]).
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===Comments===
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When $X$ is the standard euclidean space and $\mathcal{B}$ the Borel $\sigma$-algebra, the name ''singular measures'' is often used for those $\sigma$-finite measures $\mu$ which are orthogonal to the Lebesgue measure.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer (1965)</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|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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|valign="top"|{{Ref|DS}}||      N. Dunford, J.T. Schwartz, "Linear operators. General theory",    '''1''', Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}}
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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|Ha}}|| P.R. Halmos,  "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
 +
|-
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|valign="top"|{{Ref|HS}}||    E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis", Springer   (1965) {{MR|0188387}} {{ZBL|0137.03202}}
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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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|}

Revision as of 13:49, 12 August 2012

2020 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL]

If $\mu$ and $\nu$ are two $\sigma$-finite measures on the same $\sigma$-algebra $\mathcal{B}$ of subsets of $X$, then $\mu$ and $\nu$ are said to be singular (or also mutually singular, or orthogonal) if there are two sets $A,B\in\mathcal{B}$ such that $A\cap B=\emptyset$, $A\cup B = X$ and $\mu (B)=\nu (A) = 0$. The concept can be extended to signed measures or vector-valued measures: in this case it is required that $\mu (B\cap E) = \nu (A\cap E) = 0$ for every $E\in\mathcal{B}$ (cp. with Section 30 of [Ha]). The singularity of the two measures $\mu$ and $\nu$ is usually denoted by $\mu\perp\nu$.

For general, i.e. non $\sigma$-finite (nonnegative) measures, the concept can be generalized in the following way: $\mu$ and $\nu$ are singular if the only (nonnegative) measure $\alpha$ on $\mathcal{B}$ with the property \[ \alpha (A)\leq \min \{\mu (A), \nu (A)\} \qquad \forall A\in\mathcal{B} \] is the trivial measure which assigns the value $0$ to every element of $\mathcal{B}$. By the Radon-Nikodym decomposition this concept coincides with the previous one when we assume the $\sigma$-finiteness of $\mu$ and $\nu$.

Comments

When $X$ is the standard euclidean space and $\mathcal{B}$ the Borel $\sigma$-algebra, the name singular measures is often used for those $\sigma$-finite measures $\mu$ which are orthogonal to the Lebesgue measure.

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
[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 Zbl 0635.47001
[Bi] P. Billingsley, "Convergence of probability measures", Wiley (1968) MR0233396 Zbl 0172.21201
[Ha] P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[HS] E. Hewitt, K.R. Stromberg, "Real and abstract analysis", Springer (1965) MR0188387 Zbl 0137.03202
How to Cite This Entry:
Singular measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Singular_measures&oldid=27495
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article