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:

1. P ∪ N = X and P ∩ N = ∅.
2. For each E in Σ such that E ⊆ P one has μ(E) ≥ 0; that is, P is a positive set for μ.
3. For each E in Σ such that E ⊆ N one has μ(E) ≤ 0; that is, N is a negative set for μ.

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.