Probability Theory, Borel measurable, Chebyshev's inequality

Only solve in part (a),(b), please.

(Extension of the second Borel-Cantelli lemma) Let Al, A2, .. . be events in a given probability space such that _ P(A, ) = co and lim inf j.k=] P(A , nAk) n=+00 CI= P(A; )) 2 1 . (a) Show that the lim inf condition above is satisfied if the An are pairwise independent (A; and Ax are independent whenever j * k) and - P(An ) = 00.(b) Use Chebyshev's inequality to show that if In = IA , then n lim inf P * - EP(AN) > P(Ak) = 0. k=1 NIH k=16.1.5 Second Borel-Cantelli Lemma. Let ($2, , P) be a probability space, and let A1, A2, . . . be independent events in . If _, P(An ) = co, then P (lim sup,, An ) = 1. PROOF. P lim sup An ) = P NUAk OO = lim P Ak n kzn k=n = lim lim P n=+00 m-OC UAk ken Now m P JAK = P nAF = [IP(A;) by independence k=n k=n m s II exp[-P(Ap)] since P(A;) = 1 -P(Ak) k=n Do since [P(AK) = 00