Functional Analysis written by Stein and Shakarchi. Lemma4.2 on the first chapter.

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(ii) "When p=1 the argument is parallel with the above but simpler. Here we take $f_n(x)=(sign g(x))X_{E_n}(x)$,where $E_n$ is an increasing sequence of sets of finite measure whose union is X." I can not figure out how to do that.I am not smart. Can anyone know how to do that?