Hahn decomposition theorem

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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:

  1. and .
  2. For every such that , one has , i.e., is a positive set for .
  3. 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[edit]

A consequence of the Hahn decomposition theorem is the Jordan decomposition theorem, 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[edit]

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 not allowed. 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[edit]

  • 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[edit]