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.

  • $\begingroup$ What does $L^{+}$ mean? $\endgroup$ – user284331 Dec 8 '19 at 4:21
  • $\begingroup$ @user284331 I'd guess its non-negative measurable, given the context $\endgroup$ – Calvin Khor Dec 8 '19 at 4:21
  • $\begingroup$ Then my funny example goes through. $\endgroup$ – user284331 Dec 8 '19 at 4:22
  • $\begingroup$ @user284331 yes non-negative $\endgroup$ – Bob Dec 8 '19 at 4:23

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


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