Hahn decomposition theorem

In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that given a measurable space (X,Σ) and a signed measure μ defined on the σ-algebra Σ, there exist two measurable sets P and N in Σ such that:

Moreover, this decomposition is essentially unique, in the sense that for any other pair (P', N') of measurable sets fulfilling the above three conditions, the symmetric differences P Δ P' and N Δ N' are μ-null sets in the strong sense that every measurable subset of them has zero measure. The pair (P,N) is called a Hahn decomposition of the signed measure μ.

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).

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