Hahn decomposition theorem
In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space  and any signed measure  defined on the -algebra , there exist two -measurable sets,  and , of  such that:

  and  . For every  such that , one has , i.e.,  is a positive set for . For every  such that , one has , i.e.,  is a negative set for . Moreover, this decomposition is essentially unique, meaning that for any other pair  of -measurable subsets of  fulfilling the three conditions above, the symmetric differences  and  are -null sets in the strong sense that every -measurable subset of them has zero measure. The pair  is then called a Hahn decomposition of the signed measure . Jordan measure decomposition 

A consequence of the Hahn decomposition theorem is the , which states that every signed measure  defined on  has a unique decomposition into a difference  of two positive measures,  and , at least one of which is finite, such that  for every -measurable subset  and  for every -measurable subset , for any Hahn decomposition  of . We call  and  the positive and negative part of , respectively. The pair  is called a Jordan decomposition (or sometimes Hahn–Jordan decomposition) of . The two measures can be defined as

for every  and any Hahn decomposition  of . 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  of a finite signed measure , one has

for any  in . Furthermore, if  for a pair  of finite non-negative measures on , then

The last expression means that the Jordan decomposition is the minimal decomposition of  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  does not take the value  (otherwise decompose according to ). As mentioned above, a negative set is a set  such that  for every -measurable subset . Claim: Suppose that  satisfies . Then there is a negative set  such that . Proof of the claim: Define . Inductively assume for  that  has been constructed. Let

denote the supremum of  over all the -measurable subsets  of . This supremum might a priori be infinite. As the empty set  is a possible candidate for  in the definition of , and as , we have . By the definition of , there then exists a -measurable subset  satisfying

Set  to finish the induction step. Finally, define

As the sets  are disjoint subsets of , it follows from the sigma additivity of the signed measure  that

This shows that . Assume  were not a negative set. This means that there would exist a -measurable subset  that satisfies . Then  for every , so the series on the right would have to diverge to , implying that , which is a contradiction, since . Therefore,  must be a negative set. Construction of the decomposition: Set . Inductively, given , define

as the infimum of  over all the -measurable subsets  of . This infimum might a priori be . As  is a possible candidate for  in the definition of , and as , we have . Hence, there exists a -measurable subset  such that

By the claim above, there is a negative set  such that . Set  to finish the induction step. Finally, define

As the sets  are disjoint, we have for every -measurable subset  that

by the sigma additivity of . In particular, this shows that  is a negative set. Next, define . If  were not a positive set, there would exist a -measurable subset  with . Then  for all  and

which is not allowed for . Therefore,  is a positive set. Proof of the uniqueness statement:
Suppose that  is another Hahn decomposition of . Then   is a positive set and also a negative set. Therefore, every measurable subset of it has measure zero. The same applies to . As

this completes the proof. Q.E.D. References

External links
 Hahn decomposition theorem at PlanetMath. Theorems in measure theory
Articles containing proofs