View source for Hahn decomposition

{{MSC|28A15}}
[[Category:Classical measure theory]]
{{TEX|done}}
$
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\norm}[1]{\left\|#1\right\|}
$

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]]
$\mu$ on it, i.e. a $\sigma$-additive function $\mu:\mathcal{B}\to \mathbb R$. The Hahn decomposition states the existence of two nonnegative measures
$\mu^+$ 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 can therefore be interpreted as a decomposition of the space $X$. 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 decomposition 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====
{|
|-
|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}} 
|-
|valign="top"|{{Ref|Bi}}||   P. Billingsley, "Convergence of  probability measures" , Wiley (1968)   {{MR|0233396}}  {{ZBL|0172.21201}}
|-
|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}}
|-
|valign="top"|{{Ref|DS}}||   N. Dunford, J.T. Schwartz, "Linear operators. General theory" ,  '''1'''  , Interscience (1958) {{MR|0117523}} {{ZBL|0635.47001}}
|-
|valign="top"|{{Ref|Ha}}|| P.R. Halmos,   "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
|-
|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}}
|-
|}