In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space and any signed measure such that:

1. .

2. is a positive set.

3. .

Moreover, this decomposition is essentially unique fulfilling the three conditions above, the symmetric differences-null sets is then called a Hahn decomposition.

Jordan measure decomposition

A consequence of the Hahn decomposition theorem is the has a unique the positive and negative part is called a Jordan decomposition (or sometimes Hahn–Jordan decomposition. The two measures can be defined as

.

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

, one has

, then

The last expression means that the Jordan decomposition is the minimal into the 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:.

Claim:.

Proof of the claim:. Inductively has been constructed. Let

denote the supremum. This supremum might a priori satisfying

to finish the induction step. Finally, define

, it follows from the sigma additivity that

, so the series must be a negative set.

Construction of the decomposition:, define

as the infimum. This infimum might a priori such that

to finish the induction step. Finally, define

that

and

is a positive set.

Proof of the uniqueness statement:. As

this completes the proof. Q.E.D.

References

External links

• Hahn decomposition theorem at PlanetMath.