Continuity property of probabilities. (a) Let AI , A2, ...

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Continuity property of probabilities. (a) Let AI , A2, ... be an infinite sequence of events, which is "monotonically increasing," meaning that An C An+ l for every n. Let A = U;:='=IAn . Show that P(A) = limn_oo P(An). Hint: Express the event A as a union of countably many disjoint sets. (b) Suppose now that the events are "monotonically decreasing," i.e. , An+l C An for every n. Let A = n=IAn . Show that P(A) = limn_oo P(An ). Hint: Apply the result of part (a) to the complements of the events. (c) Consider a probabilistic model whose sample space is the real line. Show that p([O, (0)) = lim P([O,nj), n-oc and lim P ([n, (0)) = 0. n-oo Solution. (a) Let BI = Al and, for n 2: 2, Bn = An n A- l ' The events Bn are disjoint, and we have Uk=I Bk = An, and Uk=I Bk = A. We apply the additivity axiom to obtain 00 n P(A) = L P(Bk ) = lim P(Bk) = lim P(Uk=I Bk ) = lim P(An ). n-oc L n-oo n-oo k=l k=l (b) Let Cn = A and C = AC. Since An+l C An , we obtain Cn C Cn+l , and the events Cn are increasing. Furthermore, C = AC = (n=IAn)C = U=lA = U=lCn. Using the result from part (a) for the sequence Cn , we obtain 1 - P(A) = P(AC) = P(C) = lim P(Cn) = lim (1 - P(An)) , n-x n-oo from which we conclude that P(A) = limn_x P(An ). (c) For the first equality, use the result from part (a) with An = [O,n] and A = [0,(0). For the second, use the result from part (b) with An = [n, (0) and A = nIAn = 0.