We showed in class that there is no uniform probability distribution on the natural mumbers. There...

 

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We showed in class that there is no uniform probability distribution on the natural mumbers. There are other notions of the size of set of naturals A CN. One example is d(A) = lim sup 4u [n]| nEN where [n] = {0,1...n– 1}. Observe that d({i}) = 0 for all i and d(N) = 1. (a) Call a set A eventually periodic with period T if there is some N eN such that for all n > N, nE A if and only if n+TE A. Rigorously compute d(A) and d(A). (b) Probability measures are countably additive. Show that d is not even finitely additive by finding a set BCN such that d(B)+ d(B") # d(N). Hìnt: Think about how the different behavior of A and A° from the previous part before and after N affects the limiting argument and construct B in such a way to prevent that limiting argument from working. We showed in class that there is no uniform probability distribution on the natural mumbers. There are other notions of the size of set of naturals A CN. One example is d(A) = lim sup 4u [n]| nEN where [n] = {0,1...n– 1}. Observe that d({i}) = 0 for all i and d(N) = 1. (a) Call a set A eventually periodic with period T if there is some N eN such that for all n > N, nE A if and only if n+TE A. Rigorously compute d(A) and d(A). (b) Probability measures are countably additive. Show that d is not even finitely additive by finding a set BCN such that d(B)+ d(B") # d(N). Hìnt: Think about how the different behavior of A and A° from the previous part before and after N affects the limiting argument and construct B in such a way to prevent that limiting argument from working.

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Let Aa1a2a3aTa1a2a3aTa1a2a3 The period of this set is T W... View the full answer