Counterexamples of reverse Fatou lemma

Question

If $\{f_n\}\subset L^+$, $f\in L^+$ Is it necessarily true that $$\limsup\int f_n\leq \int \limsup f_n?$$

I know if $f$ is dominated then this result is true: Dual result of Fatou lemma

But how about counterexamples of above statement without dominated.

Answer 1

$f_{n}(x)=\dfrac{1}{n}$, $\displaystyle\int_{\mathbb{R}}f_{n}=\infty$, $\limsup_{n}f_{n}(x)=0$, $\displaystyle\int_{\mathbb{R}}\limsup_{n}f_{n}(x)=0$.