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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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2 Answers

up vote 4 down vote accepted

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

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