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Accirding to dominated convergence thm, $\lim_{n\to \infty} \int f_n = \int \lim_{n\to \infty}f_n $

if I choose f f(x)=n if 0<x<1/n and f(x)=0 o.w. then

$ \lim_{n\to\infty} \int f_n = 1, \int \lim_{n\to\infty}f_n =0 $ since $\lim_{n\to\infty}f_n $ is $0$ for almost every x . what's wrong?

user1173804
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    What is the main assumption of the dominated convergence theorem? Does it really hold in your case? – cs89 Apr 23 '23 at 09:56
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    To apply dominated convergence theorem, there is a hypothesis you have to check. You did not do that. This is a good example to use if you want to show that the hypothesis is needed. – GEdgar Apr 23 '23 at 09:56
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    The assumptions of the dominated convergence theorem are not satisfied. – Gribouillis Apr 23 '23 at 09:56
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    every f_n should be bounded by integrable function g, so you mean if I assume f_n is bounded by g, there exist anther f_N s.t. f_N>g, so f_n is not bounded by integrable function. right? – user1173804 Apr 23 '23 at 10:02

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