Hahn decomposition theorem

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Hahn decomposition theorem, named after the <a href="/facts/Austria/4qWIMIyc">Austrian</a> <a href="/facts/Mathematician/eaFSR8Fx">mathematician</a> <a href="/facts/Hans_Hahn_(mathematician)/fg9B0gCR">Hans Hahn</a>, states that for any <a href="/facts/Sigma-algebra/8LJINLNc">measurable space</a> 
 
 
 
 (
 X
 ,
 Σ
 )
 
 
 {\displaystyle (X,\Sigma )}
 
 and any <a href="/facts/Signed_measure/vukVf84A">signed measure</a> 
 
 
 
 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:

<ol><li>
 
 
 
 P
 ∪
 N
 =
 X
 
 
 {\displaystyle P\cup N=X}
 
 and 
 
 
 
 P
 ∩
 N
 =
 ∅
 
 
 {\displaystyle P\cap N=\varnothing }
 
.</li>
<li>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 <a href="/facts/Positive_and_negative_sets/W38SUGHe">positive set</a> for 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
.</li>
<li>For every 
 
 
 
 E
 ∈
 Σ
 
 
 {\displaystyle E\in \Sigma }
 
 such that 
 
 
 
 E
 ⊆
 N
 
 
 {\displaystyle E\subseteq N}
 
, one has 
 
 
 
 μ
 (
 E
 )
 ≤
 0
 
 
 {\displaystyle \mu (E)\leq 0}
 
, i.e., 
 
 
 
 N
 
 
 {\displaystyle N}
 
 is a negative set for 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
.</li></ol>
Moreover, this decomposition is <a href="/facts/Universal_property/5XJBYMNz">essentially unique</a>, 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 <a href="/facts/Symmetric_difference/W2QST1Qr">symmetric differences</a> 
 
 
 
 P
 △
 
 P
 ′
 
 
 
 {\displaystyle P\triangle P'}
 
 and 
 
 
 
 N
 △
 
 N
 ′
 
 
 
 {\displaystyle N\triangle N'}
 
 are 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
-<a href="/facts/Null_set/kMudL8Mn">null sets</a> 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 }
 
.

Hahn decomposition theorem open-in-new

Hahn decomposition theorem