What conditions imply P(F)>0P(F)>0 and P({ω})=0P(\{\omega\}) = 0 for all ω∈F\omega \in F?

Let (Ω,F,P)(\Omega, \mathcal{F},P) be a probability space. What conditions guarantee that there exists a measurable set FF such that P(F)>0P(F) > 0 and for all ω∈F\omega \in F, {ω}\{\omega \} is measurable and P({ω})=0P(\{ \omega \})=0.

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ams.org/journals/proc/1970-025-03/S0002-9939-1970-0279266-8/… Perhaps you can read that. You can always decompose your probability measure into an atomic and a nonatomic part, and you want the nonatomic part to restrict to a nonzero measure on FF. I wouldn’t know how much more you can say.
– Pedro Tamaroff♦
Oct 20 at 20:20

  

 

@PedroTamaroff Thank you for the reference.
– aduh
Oct 20 at 20:21

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