Hahn decomposition theorem

In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} and any signed measure μ {\displaystyle \mu } defined on the σ {\displaystyle \sigma } -algebra Σ {\displaystyle \Sigma } , there exist two Σ {\displaystyle \Sigma } -measurable sets, P {\displaystyle P} and N {\displaystyle N} , of X {\displaystyle X} such that: P ∪ N = X {\displaystyle P\cup N=X} and P ∩ N = ∅ {\displaystyle P\cap N=\varnothing } . For every E ∈ Σ {\displaystyle E\in \Sigma } such that E ⊆ P {\displaystyle E\subseteq P} , one has μ ( E ) ≥ 0 {\displaystyle \mu (E)\geq 0} , i.e., P {\displaystyle P} is a positive set for μ {\displaystyle \mu } .

Source: Wikipedia — Hahn decomposition theorem (CC BY-SA 4.0)

Hahn decomposition theorem

In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space ( X , Σ ) {\displaystyle (X,\Sigma )} and any signed measure μ {\displaystyle \mu } defined on the σ {\displaystyle \sigma } -algebra Σ {\displaystyle \Sigma } , there exist two Σ {\displaystyle \Sigma } -measurable sets, P {\displaystyle P} and N {\displaystyle N} , of X {\displaystyle X} such that: P ∪ N = X {\displaystyle P\cup N=X} and P ∩ N = ∅ {\displaystyle P\cap N=\varnothing } . For every E ∈ Σ {\displaystyle E\in \Sigma } such that E ⊆ P {\displaystyle E\subseteq P} , one has μ ( E ) ≥ 0 {\displaystyle \mu (E)\geq 0} , i.e., P {\displaystyle P} is a positive set for μ {\displaystyle \mu } .

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Source: Wikipedia "Hahn decomposition theorem" · CC BY-SA 4.0

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