Would you elaborate the hint for Exercise 23? I don’t quite understand what that means. Moreover, is also a bounded variant?

*[To obtain Lusin’s theorem for arbitrary measurable , first apply Lusin’s theorem to a bounded measurable function such as , , or , and figure out how to use the continuous approximation of those functions to obtain a continuous approximation to . -T]</I *

*[ may not be measurable, but the simple functions lying underneath it are. See also the general advice at https://terrytao.wordpress.com/2010/10/21/245a-problem-solving-strategies/ . -T.]*

which looks rather different from Exercise 16. Is this a contradiction?

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