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
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
As this completes the proof.