Hahn decomposition theorem

In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space ${\displaystyle (X,\Sigma )}$ and any signed measure ${\displaystyle \mu }$ defined on the ${\displaystyle \sigma }$-algebra ${\displaystyle \Sigma }$, there exist two ${\displaystyle \Sigma }$-measurable sets, ${\displaystyle P}$ and ${\displaystyle N}$, of ${\displaystyle X}$ such that:

1. ${\displaystyle P\cup N=X}$ and ${\displaystyle P\cap N=\varnothing }$.
2. For every ${\displaystyle E\in \Sigma }$ such that ${\displaystyle E\subseteq P}$, one has ${\displaystyle \mu (E)\geq 0}$, i.e., ${\displaystyle P}$ is a positive set for ${\displaystyle \mu }$.
3. For every ${\displaystyle E\in \Sigma }$ such that ${\displaystyle E\subseteq N}$, one has ${\displaystyle \mu (E)\leq 0}$, i.e., ${\displaystyle N}$ is a negative set for ${\displaystyle \mu }$.

Moreover, this decomposition is essentially unique, meaning that for any other pair ${\displaystyle (P',N')}$ of ${\displaystyle \Sigma }$-measurable subsets of ${\displaystyle X}$ fulfilling the three conditions above, the symmetric differences ${\displaystyle P\triangle P'}$ and ${\displaystyle N\triangle N'}$ are ${\displaystyle \mu }$-null sets in the strong sense that every ${\displaystyle \Sigma }$-measurable subset of them has zero measure. The pair ${\displaystyle (P,N)}$ is then called a Hahn decomposition of the signed measure ${\displaystyle \mu }$.