For n ‰¥ 1, let X_nand X be r.v.s defined on the measure space (Ω, A, μ). Then, by Theorem 4 in Chapter 3,

If and only if

 1/k = 0

for each arbitrary but fixed k = 1,2,... Replace μ by a probability measure P, and show that

if and only if P(lim sup n †’ ˆž A_n) = 0, where A_n = (|X_n| ‰¥ 1/k) for each arbitrary but fixed k = 1,2,...

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> х п—о0