(Egorov€™s Theorem). Show that, if μ is finite, thenX_n Ximplies thatX_n X.

For an arbitrary e > 0 to be kept fixed throughout and k ³ 1 integer, use Theorem 4 in order to conclude that there exists N_k = (e, k) > 0 such that μ(A_e,k) < e/2^k, k ³ 1, where A_e,k =  (|X_n €“ X| ³ 1/k). Thus, if A_e =  A_e,k, then μ (A_e) £ e. Finally, show that X_n (w) X(w) uniformly in w ÃŽ A_e.

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