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Let $\{f_n\}$ be a sequence of nonnegative functions dominated by some function $$ g \in L^1. $$ Then, the reverse Fatou lemma says $$ \limsup \int f_n \le \int \limsup f_n. $$

Is it possible to give an example where the inequality is strict?

I tried functions like $$ f_n=\chi_{(n,n+1)}, $$ but the dominating function is not integrble.

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$$f_{2n}=\chi_{[0,1]}\qquad f_{2n+1}=\chi_{[1,2]}\qquad g=\chi_{[0,2]}$$

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  • $\begingroup$ Can you explain how this gives the answer? Don’t we end up with $\int$ lim sup $f_n(x) = 0$? $\endgroup$ – TuringTester69 Nov 4 '18 at 22:10
  • $\begingroup$ @JaneDoe No we do not end up with what you say. What is $\limsup f_n$ according to you? $\endgroup$ – Did Nov 5 '18 at 7:18
  • $\begingroup$ Sorry, maybe I misunderstood. Are we taking the integral over $\mathbb{R}$ or over $[0,2]$? If we are taking it over the reals, I think $\limsup f_n$ should be 0 because $f_n (x) = 0$ $\forall x > 2$. However, if it is over $[0,2]$ then I think $\limsup f_n$ should be 1. $\endgroup$ – TuringTester69 Nov 6 '18 at 1:11
  • $\begingroup$ @JaneDoe You seem to have in mind $$\limsup_{x\to\infty}f_n(x)$$ for some unspecified value of $n$, that is, a limsup which is irrelevant, instead of, for each given $x$, $$h(x)=\limsup_{n\to\infty}f_n(x)$$ which defines the function of interest $$h=\limsup_{n\to\infty}f_n$$ $\endgroup$ – Did Nov 6 '18 at 16:03
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What about $f_n(x)=x(-1)^n \chi_{[-1,1]}(x)+\chi_{[-1,1]}, ~g=2\chi_{[-1,1]}$?

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    $\begingroup$ nonnegative functions. $\endgroup$ – Did Apr 29 '13 at 16:58
  • $\begingroup$ sorry, but adding $1$ should fix that $\endgroup$ – Julian Apr 29 '13 at 17:01

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