# Counterexamples of reverse Fatou lemma

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.

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

## 1 Answer

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