Difference between revisions of "Hahn decomposition"
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| − | A concept in classical measure theory related to the [[Jordan decomposition]] | + | 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]] |
$\mu$ on it, i.e. a $\sigma$-additive function $\mu:\mathcal{B}\to \mathbb R$. The Jordan decomposition states the existence of two nonnegative measures | $\mu$ 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 is the decomposition of | + | 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. | are uniquely determined by the property given above, the sets $X^+$ and $X^-$ are not. | ||
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\begin{align*} | \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\}\\ | ||
| − | \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*} | \end{align*} | ||
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 |
Hahn decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hahn_decomposition&oldid=27692