Positive and negative sets

In <a href="/facts/Measure_theory/yCq7zlI4">measure theory</a>, given a <a href="/facts/Sigma-algebra/8LJINLNc">measurable space</a> 
 
 
 
 (
 X
 ,
 Σ
 )
 
 
 {\displaystyle (X,\Sigma )}
 
 and a <a href="/facts/Signed_measure/vukVf84A">signed measure</a> 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 on it, a set 
 
 
 
 A
 ∈
 Σ
 
 
 {\displaystyle A\in \Sigma }
 
 is called a positive set for 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 if every 
 
 
 
 Σ
 
 
 {\displaystyle \Sigma }
 
-measurable subset of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 has nonnegative measure; that is, for every 
 
 
 
 E
 ⊆
 A
 
 
 {\displaystyle E\subseteq A}
 
 that satisfies 
 
 
 
 E
 ∈
 Σ
 ,
 
 
 {\displaystyle E\in \Sigma ,}
 
 
 
 
 
 μ
 (
 E
 )
 ≥
 0
 
 
 {\displaystyle \mu (E)\geq 0}
 
 holds. 
Similarly, a set 
 
 
 
 A
 ∈
 Σ
 
 
 {\displaystyle A\in \Sigma }
 
 is called a negative set for 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 if for every subset 
 
 
 
 E
 ⊆
 A
 
 
 {\displaystyle E\subseteq A}
 
 satisfying 
 
 
 
 E
 ∈
 Σ
 ,
 
 
 {\displaystyle E\in \Sigma ,}
 
 
 
 
 
 μ
 (
 E
 )
 ≤
 0
 
 
 {\displaystyle \mu (E)\leq 0}
 
 holds.
Intuitively, a measurable set 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is positive (resp. negative) for 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 if 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is nonnegative (resp. nonpositive) everywhere on 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
 Of course, if 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 is a <a href="/facts/Measure_(mathematics)/yCq7zlI4">nonnegative measure</a>, every element of 
 
 
 
 Σ
 
 
 {\displaystyle \Sigma }
 
 is a positive set for 
 
 
 
 μ
 .
 
 
 {\displaystyle \mu .}

In the light of <a href="/facts/Radon%E2%80%93Nikodym_theorem/uWepNG2B">Radon–Nikodym theorem</a>, if 
 
 
 
 ν
 
 
 {\displaystyle \nu }
 
 is a σ-finite positive measure such that 
 
 
 
 
 |
 
 μ
 
 |
 
 ≪
 ν
 ,
 
 
 {\displaystyle |\mu |\ll \nu ,}
 
 a set 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is a positive set for 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the Radon–Nikodym derivative 
 
 
 
 d
 μ
 
 /
 
 d
 ν
 
 
 {\displaystyle d\mu /d\nu }
 
 is nonnegative 
 
 
 
 ν
 
 
 {\displaystyle \nu }
 
-almost everywhere on 
 
 
 
 A
 .
 
 
 {\displaystyle A.}
 
 Similarly, a negative set is a set where 
 
 
 
 d
 μ
 
 /
 
 d
 ν
 ≤
 0
 
 
 {\displaystyle d\mu /d\nu \leq 0}
 
 
 
 
 
 ν
 
 
 {\displaystyle \nu }
 
-almost everywhere.

Positive and negative sets open-in-new

Positive and negative sets