If $f_n \rightarrow f$ pointwise, then does $$\liminf \int f_n=\lim\int f_n?$$
I know that $\liminf f_n=\lim f_n$ since the sequence converges, but I'm not sure if the $(L)$ integral throws us off.
I'm trying to prove the Fatou's Reverse Lemma, and I got stuck.
EDITED: $f$ is integrable and $f_n\le f$
Best Answer
If we had some guarantee that the limit on the right hand side exists, then of course this equality would be true, because for a convergent sequence $a_n = \int f_n d \mu$ we would have $\liminf a_n = \lim a_n$.
But in your case there doesn't seem to be such a guarantee. For example, consider a sequence of functions $f_n: \mathbb{R} \to \mathbb{R}$ like this: for $n$ even, $f_n=0$. And for $n$ odd, $f_n = 1_{[n, n+1]}$. Then $\int f_n d\mu$ is $0$ for n even and $1$ for n odd, so $\liminf$ on the LHS is equal to $0$, and the $\lim$ on the RHS doesn't exist.