Posted by J. Holmer on November 14, 2007, 10:57 am, in reply to "HW exercise 18"
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No assumptions on boundedness or finite measure are necessary in this problem. First appeal to the result that any measurable function is the a.e. limit of a sequence of step functions f_n. This statement is true even if f takes on values +\infty or -\infty and even if it is nonzero on a set of infinite measure. For each step function f_n, show that there is a sequence of continuous functions f_{n,k} so that f_{n,k} \to f_n a.e. as k \to \infty. Then argue that for each n there is k_n such that f_{n,k_n} converges a.e. to f. This last part is proved using Borel-Cantelli.
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