# Strict inequality in Reverse Fatou lemma: $\varlimsup \int f_n\le \int \varlimsup f_n$

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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$$f_{2n}=\chi_{[0,1]}\qquad f_{2n+1}=\chi_{[1,2]}\qquad g=\chi_{[0,2]}$$
What about $f_n(x)=x(-1)^n \chi_{[-1,1]}(x)+\chi_{[-1,1]}, ~g=2\chi_{[-1,1]}$?
nonnegative functions. –  Did Apr 29 '13 at 16:58
sorry, but adding $1$ should fix that –  Julian Apr 29 '13 at 17:01