Ph.D. Qualifying Exam – Spring 2004

∫µ(A 1 ) δ 1 and |f k1 | dµ > ǫA 1Let’s rename f k1 to g 1 . This is the first step in the inductive construction of a subsequence(g n ) n∈N of (f n ) n∈N , a sequence (A n ) n∈N of measurable sets, and a sequence (δ n ) n∈N ofpositive real numbers.Suppose that we have constructed the first terms g 1 , . . .,g n of the subsequence, setsA 1 , . . ., A n and positive real numbers δ 1 , . . .,δ n , with the following properties:• For m n, δ m is such that∀A ∈ A(µ(A) δm =⇒• For m n, A m and g m are such thatµ(A m ) δ m∫Aand|g j | dµ < ǫ for j = 1, . . .,m − 1)2m ∫A m|g m | dµ > ǫUsing Theorem 1 and (⋆), it is clear that we can construct δ n+1 , A n+1 and g n+1 . So thetwo properties listed above are actually satisfied for every n. Hope this is clear enough...For every positive integer n, letThe sets (B n ) n∈N are pairwise disjointB n = A n \ ( n−1 ⋃j=1A j)4