Necessarily the case that $\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx$?

Question

Let $f_n$ be a sequence of nonnegative Lebesgue measurable functions on $\mathbb{R}$. Is it necessarily true that$$\limsup_{n \to \infty} \int f_n\,dx \le \int \limsup_{n \to \infty} f_n\,dx?$$

Answer 1

It is not. In particular, take $f_n(x) = 1_{[n,n+1]}$. Then the left side comes out to $1$, but the right comes out to $0$.

Answer 2

No, for $$f_n:=\mathbf 1_{[n,n+1]}$$ we have $\int f_n=1$ for all $n\in\Bbb N$ so $$\lim \sup \int f_n=1.$$ However $\lim \sup f_n=0$ thus $$ \int\lim \sup f_n=0.$$