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: Moreover, this decomposition is essentially unique, meaning that for any other pair
fulfilling the three conditions above, the symmetric differences
-null sets in the strong sense that every
is then called a Hahn decomposition of the signed measure
A consequence of the Hahn decomposition theorem is the Jordan decomposition theorem, which states that every signed measure
has a unique decomposition into a difference
the positive and negative part of
The two measures can be defined as for every
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
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 claim: Define
This supremum might a priori be infinite.
to finish the induction step.
Finally, define As the sets
, it follows from the sigma additivity of the signed measure
Construction of the decomposition: Set
such that By the claim above, there is a negative set
to finish the induction step.
Finally, define As the sets
were not a positive set, there would exist a
and[clarification needed] which is not allowed for
Proof of the uniqueness statement: Suppose that