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A partition of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h0461401.png" />, on which a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h0461402.png" />-additive set function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h0461403.png" /> is given on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h0461404.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h0461405.png" /> of subsets, into two subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h0461406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h0461407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h0461408.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h0461409.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h04614010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h04614011.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h04614012.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h04614013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h04614014.png" />. Such a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046140/h04614015.png" /> is not unique, in general.
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{{MSC|28A15}}
 
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[[Category:Classical measure theory]]
====References====
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{{TEX|done}}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,  J.T. Schwartz,  "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR></table>
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$
  
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A concept in classical measure theory related to the [[Jordan decomposition]]. Consider a [[Algebra of sets|σ-algebra]] $\mathcal{B}$ of subsets of a set $X$ and a [[Signed measure|signed measure]]
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$\mu$ on it, i.e. a $\sigma$-additive function $\mu:\mathcal{B}\to \mathbb R$. The Jordan decomposition states the existence of two nonnegative measures
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$\mu^+$ and $\mu^-$ which are mutually singular (see [[Absolute continuity]]) and such that $\mu =\mu^+-\mu^-$. The property of being mutually singular
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translates into the existence of a set $X^+\in\mathcal{B}$ such that $\mu^+ (X\setminus X^+)=0$ and $\mu^- (X^+)=0$ (see Section 29 of {{Cite|Ha}}). If we denote by $X^-$ the complement of
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$X^+$, we then conclude that $\mu (A)\geq 0$ for any $A\in\mathcal{B}$ with $A\subset X^+$ and $\mu (A)\leq 0$ for any $A\in\mathcal{B}$ with $A\subset X^-$.
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The Hahn decomposition is the decomposition of $X$ into the subsets $X^+$ and $X^-$. Observe however that, while the two measures $\mu^+$ and $\mu^-$
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are uniquely determined by the property given above, the sets $X^+$ and $X^-$ are not.
  
====Comments====
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The Hahn and the Jordan decompositions can be derived as a corollary of the [[Radon-Nikodym theorem]] (applied to $\mu$ and its total variation,
See also [[Jordan decomposition|Jordan decomposition]]. Instead of Hahn decomposition the phrase Hahn–Jordan decomposition is also used.
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see [[Signed measure]]), or can be proved directly by setting
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\begin{align*}
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\mu^+ (B) &= \sup \{ \mu (A): A\in \mathcal{B}, A\subset B\}\\
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\mu^- (B) &= \sup \{ -\mu (A): A\in \mathcal{B}, A\subset B\}\, .
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\end{align*}
  
 
====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">  W. Rudin,   "Real and complex analysis" , McGraw-Hill (1966) pp. 57</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.L. Royden, [[Royden, "Real analysis"|"Real analysis"]], Macmillan (1968)</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}} {{ZBL|0635.47001}}
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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, Cambridge1995. {{MR|1333890}} {{ZBL|0911.28005}}
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|-
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|}

Latest revision as of 10:49, 10 December 2012

2020 Mathematics Subject Classification: Primary: 28A15 [MSN][ZBL] $ \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\norm}[1]{\left\|#1\right\|} $

A concept in classical measure theory related to the Jordan decomposition. Consider a σ-algebra $\mathcal{B}$ of subsets of a set $X$ and a signed measure $\mu$ on it, i.e. a $\sigma$-additive function $\mu:\mathcal{B}\to \mathbb R$. The Jordan decomposition states the existence of two nonnegative measures $\mu^+$ and $\mu^-$ which are mutually singular (see Absolute continuity) and such that $\mu =\mu^+-\mu^-$. The property of being mutually singular translates into the existence of a set $X^+\in\mathcal{B}$ such that $\mu^+ (X\setminus X^+)=0$ and $\mu^- (X^+)=0$ (see Section 29 of [Ha]). If we denote by $X^-$ the complement of $X^+$, we then conclude that $\mu (A)\geq 0$ for any $A\in\mathcal{B}$ with $A\subset X^+$ and $\mu (A)\leq 0$ for any $A\in\mathcal{B}$ with $A\subset X^-$. The Hahn decomposition is the decomposition of $X$ into the subsets $X^+$ and $X^-$. Observe however that, while the two measures $\mu^+$ and $\mu^-$ are uniquely determined by the property given above, the sets $X^+$ and $X^-$ are not.

The Hahn and the Jordan decompositions can be derived as a corollary of the Radon-Nikodym theorem (applied to $\mu$ and its total variation, see Signed measure), or can be proved directly by setting \begin{align*} \mu^+ (B) &= \sup \{ \mu (A): A\in \mathcal{B}, A\subset B\}\\ \mu^- (B) &= \sup \{ -\mu (A): A\in \mathcal{B}, A\subset B\}\, . \end{align*}

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 Zbl 0635.47001
[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:
Hahn decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hahn_decomposition&oldid=25514
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article