Hello everybody,
I need to prove the following theorem:
"Given a σ-finite premeasure u in a ring R ⊆ P(S), where S is an arbitrary non-empty set.
If A ⊆ S is u*-measurable (A satisfies the Carathéodory condition), then there is a N ⊆ S
such that u*(N)=0 and AUN ∈ σ(R)."
Does anyone have an idea how to prove this (in a simple way)?
PS: Since I'm not American I don't know if the notation I used is common:
- u* is the outer measure induced by the premeasure u.
- σ(R) is the smallest σ-Algebra containing R.[FONT=MathJax_Main]
[/FONT]
I need to prove the following theorem:
"Given a σ-finite premeasure u in a ring R ⊆ P(S), where S is an arbitrary non-empty set.
If A ⊆ S is u*-measurable (A satisfies the Carathéodory condition), then there is a N ⊆ S
such that u*(N)=0 and AUN ∈ σ(R)."
Does anyone have an idea how to prove this (in a simple way)?
PS: Since I'm not American I don't know if the notation I used is common:
- u* is the outer measure induced by the premeasure u.
- σ(R) is the smallest σ-Algebra containing R.[FONT=MathJax_Main]
[/FONT]
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