No generalization of subadditivity for limit superior like Fatou's lemma.

Question

We know that if $(a_n)$ and $(b_n)$ are two sequences in $\overline{\mathbb R}$, then we have $\limsup(a_n+b_n)\leq \limsup a_n +\limsup b_n$,So if $\{f_n:\mathbb N\to \overline{\mathbb R}\}$ is a sequence of functions,then for any $N\in \mathbb N$ we have, $\limsup\limits_{n\to \infty}\sum\limits_{k=1}^N f_n(k)\leq \sum\limits_{k=1}^N\limsup\limits_{n\to \infty}f_n(k)$.But this cannot be generalized to $\limsup\int_X f_nd\mu\leq \int_X\limsup f_n d\mu$ even when $f_n\geq 0$ for all $n\in \mathbb N$.I think the proof will fail where we use monotone convergence theorem which is valid for monotone increasing sequence but not for decreasing ones.So,I can take $f_n=\chi_{[n,\infty)}$ with respect to the measure space $\mathbb R$ with Lebesgue measure $\lambda$,then $\limsup\int_{\mathbb R} f_nd\lambda=+\infty$ but $\int_{\mathbb R} \limsup f_n d\lambda=0$.So,unlike Fatou's lemma there is no generalization for limit superior.I want to know whether I can put some extra conditions so that this is true?Can someone help me?

Answer 1

If there is an intergable function $f$ such that $|f_n| \leq f$ for all $n$ then application of Fatou's Lemma to $f-f_n$ gives $\int \lim \inf (f-f_n) \leq \lim \inf \int (f-f_n)$ and this is equivalent to $\int \lim \sup f_n \geq \lim \sup \int f_n$.