Hahn decomposition

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

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