View source for Jordan decomposition (of a signed measure)

{{MSC|28A15}}
[[Category:Classical measure theory]]
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A  concept in classical measure theory related to the [[Hahn  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
$\mu^+$ 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
$X$, 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====
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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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|valign="top"|{{Ref|Bi}}||    P. Billingsley, "Convergence of  probability measures" , Wiley (1968)    {{MR|0233396}}  {{ZBL|0172.21201}}
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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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|valign="top"|{{Ref|Ha}}|| P.R. Halmos,   "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
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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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