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: 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: Moreover, this decomposition is essentially unique, meaning that for any other pair ( P ′ , N ′ ) {displaystyle (P',N')} of Σ {displaystyle Sigma } -measurable subsets of X {displaystyle X} fulfilling the three conditions above, the symmetric differences P △ P ′ {displaystyle P riangle P'} and N △ N ′ {displaystyle N riangle N'} are μ {displaystyle mu } -null sets in the strong sense that every Σ {displaystyle Sigma } -measurable subset of them has zero measure. The pair ( P , N ) {displaystyle (P,N)} is then called a Hahn decomposition of the signed measure μ {displaystyle mu } . A consequence of the Hahn decomposition theorem is the Jordan decomposition theorem, which states that every signed measure μ {displaystyle mu } defined on Σ {displaystyle Sigma } has a unique decomposition into a difference μ = μ + − μ − {displaystyle mu =mu ^{+}-mu ^{-}} of two positive measures, μ + {displaystyle mu ^{+}} and μ − {displaystyle mu ^{-}} , at least one of which is finite, such that μ + ( E ) = 0 {displaystyle {mu ^{+}}(E)=0} for every Σ {displaystyle Sigma } -measurable subset E ⊆ N {displaystyle Esubseteq N} and μ − ( E ) = 0 {displaystyle {mu ^{-}}(E)=0} for every Σ {displaystyle Sigma } -measurable subset E ⊆ P {displaystyle Esubseteq P} , for any Hahn decomposition ( P , N ) {displaystyle (P,N)} of μ {displaystyle mu } . We call μ + {displaystyle mu ^{+}} and μ − {displaystyle mu ^{-}} the positive and negative part of μ {displaystyle mu } , respectively. The pair ( μ + , μ − ) {displaystyle (mu ^{+},mu ^{-})} is called a Jordan decomposition (or sometimes Hahn–Jordan decomposition) of μ {displaystyle mu } . The two measures can be defined as for every E ∈ Σ {displaystyle Ein Sigma } and any Hahn decomposition ( P , N ) {displaystyle (P,N)} of μ {displaystyle mu } .