Monday, 15 September 2014

real analysis - Additive but not sigma-additive function





Give an example of a measure space (Ω,F) and a function μ on F that is additive but not σ-additive, i.e. μ(Ai)=μ(Ai) for a finite collection of disjoint Ai but not for some infinite collections.




I know a measure function defined on σ-algebra is σ-additive, but I struggle finding a function that would not be additive for infinite collections. Can someone give me an example and show me why?


Answer



Hint Consider Ω=N, the power set F=P(N) and the mapping μ:F[0,], μ(A):={0,A is a finite set,otherwise.


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