Hello everyone! I need to solve the following problem but I can't do it.
Let P_{1} and P_{2} be two sigma-additive over the set N of natural numbers probabilities such that P_{1}([0,n])>=P_{2}([0,2]) for all n. I know that for sure there exist n_{1} and n_{2} such that forall c>0, forall n>=n_{2} E_{P2}[(x+c)1_{[0,n]}]>E_{P2}[x] and forall n>=n_{1 }E_{P1}[(x+c)1_{[0,n]}]>E_{P1}[x]. With x a non-negative, real-valued, bounded sequence.
I want to prove that P_{1}([0,n])>=P_{2}([0,2]) implies that whenever for an n E_{P2}[(x+c)1_{[0,n]}]>E_{P2}[x] then for the same n E_{P1}[(x+c)1_{[0,n]}]>E_{P1}[x].
Thanks a lot!