Hahn decomposition theorem

In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space ${\displaystyle (X,\Sigma )}$ and any signed measure ${\displaystyle \mu }$ defined on the ${\displaystyle \sigma }$-algebra ${\displaystyle \Sigma }$, there exist two ${\displaystyle \Sigma }$-measurable sets, ${\displaystyle P}$ and ${\displaystyle N}$, of ${\displaystyle X}$ such that:

1. ${\displaystyle P\cup N=X}$ and ${\displaystyle P\cap N=\varnothing }$.
2. For every ${\displaystyle E\in \Sigma }$ such that ${\displaystyle E\subseteq P}$, one has ${\displaystyle \mu (E)\geq 0}$, i.e., ${\displaystyle P}$ is a positive set for ${\displaystyle \mu }$.
3. For every ${\displaystyle E\in \Sigma }$ such that ${\displaystyle E\subseteq N}$, one has ${\displaystyle \mu (E)\leq 0}$, i.e., ${\displaystyle N}$ is a negative set for ${\displaystyle \mu }$.

Moreover, this decomposition is essentially unique, meaning that for any other pair ${\displaystyle (P',N')}$ of ${\displaystyle \Sigma }$-measurable subsets of ${\displaystyle X}$ fulfilling the three conditions above, the symmetric differences ${\displaystyle P\triangle P'}$ and ${\displaystyle N\triangle N'}$ are ${\displaystyle \mu }$-null sets in the strong sense that every ${\displaystyle \Sigma }$-measurable subset of them has zero measure. The pair ${\displaystyle (P,N)}$ is then called a Hahn decomposition of the signed measure ${\displaystyle \mu }$.

Jordan measure decomposition

A consequence of the Hahn decomposition theorem is the Jordan decomposition theorem, which states that every signed measure ${\displaystyle \mu }$  defined on ${\displaystyle \Sigma }$  has a unique decomposition into a difference ${\displaystyle \mu =\mu ^{+}-\mu ^{-}}$  of two positive measures, ${\displaystyle \mu ^{+}}$  and ${\displaystyle \mu ^{-}}$ , at least one of which is finite, such that ${\displaystyle {\mu ^{+}}(E)=0}$  for every ${\displaystyle \Sigma }$ -measurable subset ${\displaystyle E\subseteq N}$  and ${\displaystyle {\mu ^{-}}(E)=0}$  for every ${\displaystyle \Sigma }$ -measurable subset ${\displaystyle E\subseteq P}$ , for any Hahn decomposition ${\displaystyle (P,N)}$  of ${\displaystyle \mu }$ . We call ${\displaystyle \mu ^{+}}$  and ${\displaystyle \mu ^{-}}$  the positive and negative part of ${\displaystyle \mu }$ , respectively. The pair ${\displaystyle (\mu ^{+},\mu ^{-})}$  is called a Jordan decomposition (or sometimes Hahn–Jordan decomposition) of ${\displaystyle \mu }$ . The two measures can be defined as

${\displaystyle {\mu ^{+}}(E):=\mu (E\cap P)\qquad {\text{and}}\qquad {\mu ^{-}}(E):=-\mu (E\cap N)}$ 

for every ${\displaystyle E\in \Sigma }$  and any Hahn decomposition ${\displaystyle (P,N)}$  of ${\displaystyle \mu }$ .

Note that the Jordan decomposition is unique, while the Hahn decomposition is only essentially unique.

The Jordan decomposition has the following corollary: Given a Jordan decomposition ${\displaystyle (\mu ^{+},\mu ^{-})}$  of a finite signed measure ${\displaystyle \mu }$ , one has

${\displaystyle {\mu ^{+}}(E)=\sup _{B\in \Sigma ,~B\subseteq E}\mu (B)\quad {\text{and}}\quad {\mu ^{-}}(E)=-\inf _{B\in \Sigma ,~B\subseteq E}\mu (B)}$ 

for any ${\displaystyle E}$  in ${\displaystyle \Sigma }$ . Furthermore, if ${\displaystyle \mu =\nu ^{+}-\nu ^{-}}$  for a pair ${\displaystyle (\nu ^{+},\nu ^{-})}$  of finite non-negative measures on ${\displaystyle X}$ , then

${\displaystyle \nu ^{+}\geq \mu ^{+}\quad {\text{and}}\quad \nu ^{-}\geq \mu ^{-}.}$ 

The last expression means that the Jordan decomposition is the minimal decomposition of ${\displaystyle \mu }$  into a difference of non-negative measures. This is the minimality property of the Jordan decomposition.

Proof of the Jordan decomposition: For an elementary proof of the existence, uniqueness, and minimality of the Jordan measure decomposition see Fischer (2012).

Proof of the Hahn decomposition theorem

Preparation: Assume that ${\displaystyle \mu }$  does not take the value ${\displaystyle -\infty }$  (otherwise decompose according to ${\displaystyle -\mu }$ ). As mentioned above, a negative set is a set ${\displaystyle A\in \Sigma }$  such that ${\displaystyle \mu (B)\leq 0}$  for every ${\displaystyle \Sigma }$ -measurable subset ${\displaystyle B\subseteq A}$ .

Claim: Suppose that ${\displaystyle D\in \Sigma }$  satisfies ${\displaystyle \mu (D)\leq 0}$ . Then there is a negative set ${\displaystyle A\subseteq D}$  such that ${\displaystyle \mu (A)\leq \mu (D)}$ .

Proof of the claim: Define ${\displaystyle A_{0}:=D}$ . Inductively assume for ${\displaystyle n\in \mathbb {N} _{0}}$  that ${\displaystyle A_{n}\subseteq D}$  has been constructed. Let

${\displaystyle t_{n}:=\sup(\{\mu (B)\mid B\in \Sigma ~{\text{and}}~B\subseteq A_{n}\})}$ 

denote the supremum of ${\displaystyle \mu (B)}$  over all the ${\displaystyle \Sigma }$ -measurable subsets ${\displaystyle B}$  of ${\displaystyle A_{n}}$ . This supremum might a priori be infinite. As the empty set ${\displaystyle \varnothing }$  is a possible candidate for ${\displaystyle B}$  in the definition of ${\displaystyle t_{n}}$ , and as ${\displaystyle \mu (\varnothing )=0}$ , we have ${\displaystyle t_{n}\geq 0}$ . By the definition of ${\displaystyle t_{n}}$ , there then exists a ${\displaystyle \Sigma }$ -measurable subset ${\displaystyle B_{n}\subseteq A_{n}}$  satisfying

${\displaystyle \mu (B_{n})\geq \min \!\left(1,{\frac {t_{n}}{2}}\right).}$ 

Set ${\displaystyle A_{n+1}:=A_{n}\setminus B_{n}}$  to finish the induction step. Finally, define

${\displaystyle A:=D{\Bigg \backslash }\bigcup _{n=0}^{\infty }B_{n}.}$ 

As the sets ${\displaystyle (B_{n})_{n=0}^{\infty }}$  are disjoint subsets of ${\displaystyle D}$ , it follows from the sigma additivity of the signed measure ${\displaystyle \mu }$  that

${\displaystyle \mu (D)=\mu (A)+\sum _{n=0}^{\infty }\mu (B_{n})\geq \mu (A)+\sum _{n=0}^{\infty }\min \!\left(1,{\frac {t_{n}}{2}}\right)\geq \mu (A).}$ 

This shows that ${\displaystyle \mu (A)\leq \mu (D)}$ . Assume ${\displaystyle A}$  were not a negative set. This means that there would exist a ${\displaystyle \Sigma }$ -measurable subset ${\displaystyle B\subseteq A}$  that satisfies ${\displaystyle \mu (B)>0}$ . Then ${\displaystyle t_{n}\geq \mu (B)}$  for every ${\displaystyle n\in \mathbb {N} _{0}}$ , so the series on the right would have to diverge to ${\displaystyle +\infty }$ , implying that ${\displaystyle \mu (D)=+\infty }$ , which is a contradiction, since ${\displaystyle \mu (D)\leq 0}$ . Therefore, ${\displaystyle A}$  must be a negative set.

Construction of the decomposition: Set ${\displaystyle N_{0}=\varnothing }$ . Inductively, given ${\displaystyle N_{n}}$ , define

${\displaystyle s_{n}:=\inf(\{\mu (D)\mid D\in \Sigma ~{\text{and}}~D\subseteq X\setminus N_{n}\}).}$ 

as the infimum of ${\displaystyle \mu (D)}$  over all the ${\displaystyle \Sigma }$ -measurable subsets ${\displaystyle D}$  of ${\displaystyle X\setminus N_{n}}$ . This infimum might a priori be ${\displaystyle -\infty }$ . As ${\displaystyle \varnothing }$  is a possible candidate for ${\displaystyle D}$  in the definition of ${\displaystyle s_{n}}$ , and as ${\displaystyle \mu (\varnothing )=0}$ , we have ${\displaystyle s_{n}\leq 0}$ . Hence, there exists a ${\displaystyle \Sigma }$ -measurable subset ${\displaystyle D_{n}\subseteq X\setminus N_{n}}$  such that

${\displaystyle \mu (D_{n})\leq \max \!\left({\frac {s_{n}}{2}},-1\right)\leq 0.}$ 

By the claim above, there is a negative set ${\displaystyle A_{n}\subseteq D_{n}}$  such that ${\displaystyle \mu (A_{n})\leq \mu (D_{n})}$ . Set ${\displaystyle N_{n+1}:=N_{n}\cup A_{n}}$  to finish the induction step. Finally, define

${\displaystyle N:=\bigcup _{n=0}^{\infty }A_{n}.}$ 

As the sets ${\displaystyle (A_{n})_{n=0}^{\infty }}$  are disjoint, we have for every ${\displaystyle \Sigma }$ -measurable subset ${\displaystyle B\subseteq N}$  that

${\displaystyle \mu (B)=\sum _{n=0}^{\infty }\mu (B\cap A_{n})}$ 

by the sigma additivity of ${\displaystyle \mu }$ . In particular, this shows that ${\displaystyle N}$  is a negative set. Next, define ${\displaystyle P:=X\setminus N}$ . If ${\displaystyle P}$  were not a positive set, there would exist a ${\displaystyle \Sigma }$ -measurable subset ${\displaystyle D\subseteq P}$  with ${\displaystyle \mu (D)<0}$ . Then ${\displaystyle s_{n}\leq \mu (D)}$  for all ${\displaystyle n\in \mathbb {N} _{0}}$  and^{[clarification needed]}

${\displaystyle \mu (N)=\sum _{n=0}^{\infty }\mu (A_{n})\leq \sum _{n=0}^{\infty }\max \!\left({\frac {s_{n}}{2}},-1\right)=-\infty ,}$ 

which is not allowed for ${\displaystyle \mu }$ . Therefore, ${\displaystyle P}$  is a positive set.

Proof of the uniqueness statement: Suppose that ${\displaystyle (N',P')}$  is another Hahn decomposition of ${\displaystyle X}$ . Then ${\displaystyle P\cap N'}$  is a positive set and also a negative set. Therefore, every measurable subset of it has measure zero. The same applies to ${\displaystyle N\cap P'}$ . As

${\displaystyle P\triangle P'=N\triangle N'=(P\cap N')\cup (N\cap P'),}$ 

this completes the proof. Q.E.D.

References

• Billingsley, Patrick (1995). Probability and Measure -- Third Edition. Wiley Series in Probability and Mathematical Statistics. New York: John Wiley & Sons. ISBN 0-471-00710-2.
• Fischer, Tom (2012). "Existence, uniqueness, and minimality of the Jordan measure decomposition". arXiv:1206.5449 [math.ST].

External links

• Hahn decomposition theorem at PlanetMath.
• "Hahn decomposition", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "Jordan decomposition (of a signed measure)", Encyclopedia of Mathematics, EMS Press, 2001 [1994]