Hahn Decomposition Theorem - Jordan Measure Decomposition

Jordan Measure Decomposition

A consequence of the Hahn decomposition theorem is the Jordan decomposition theorem, which states that every signed measure μ has a unique decomposition into a difference μ = μ+ − μ– of two positive measures μ+ and μ–, at least one of which is finite, such that μ+(E) = 0 if E ⊆ N and μ−(E) = 0 if E ⊆ P for any Hahn decomposition (P,N) of μ. μ+ and μ– are called the positive and negative part of μ, respectively. The pair (μ+, μ–) is called a Jordan decomposition (or sometimes Hahn–Jordan decomposition) of μ. The two measures can be defined as

and

for every E in Σ and any Hahn decomposition (P,N) of μ.

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 (μ+, μ−) of a finite signed measure μ,

and

for any E in Σ. Also, if μ = ν+ − ν– for a pair of finite non-negative measures (ν+, ν–), then

The last expression means that the Jordan decomposition is the minimal decomposition of μ 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).