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Difference between revisions of "Hahn decomposition"

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A concept in classical measure theory, also called [[Jordan decomposition]] by some authors. Consider a [[Algebra of sets|σ-algebra]] $\mathcal{B}$ of subsets of a set $X$ and a [[Signed measure|signed measure]]
+
A concept in classical measure theory related to the [[Jordan decomposition]] by some authors. Consider a [[Algebra of sets|σ-algebra]] $\mathcal{B}$ of subsets of a set $X$ and a [[Signed measure|signed measure]]
$\mu$ on it, i.e. a $\sigma$-additive function $\mu:\mathcal{B}\to \mathbb R$. The Hahn decomposition states the existence of two nonnegative measures
+
$\mu$ 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
 
$\mu^+$ 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 {{Cite|Ha}}). If we denote by $X^-$ the complement of
 
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
 
$X$, 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 $A\subset X^-$.
 
$X$, 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 can therefore be interpreted as a decomposition of the space $X$. Observe however that, while the two measures $\mu^+$ and $\mu^-$
+
The Hahn decomposition is the decomposition of the $X$ 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.
 
are uniquely determined by the property given above, the sets $X^+$ and $X^-$ are not.
  
The Hahn decomposition can be derived as a corollary of the [[Radon-Nikodym theorem]] (applied to $\mu$ and its total variation,
+
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
 
see [[Signed measure]]), or can be proved directly by setting
 
\begin{align*}
 
\begin{align*}

Revision as of 14:06, 20 August 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 by some authors. 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 the $X$ 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=27646
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article