[Math] Strict inequality for Fatou’s lemma

Question

I'm interested in knowing whether there is a condition for general measure spaces under which we know that we can only achieve the strict inequality of Fatou's lemma. I am working in the situation that $f_n \rightarrow f$, and the limit of the integrals do exist so that Fatou's lemma says
$$ \int f \leq \lim_{n \rightarrow \infty} \int f_n \;. $$
Is there a condition on $f_n$ and $f$ which ensures
$$ \int f < \lim_{n \rightarrow \infty} \int f_n \;. $$

Answer 1

There is a nice discussion of this point in ANALYSIS by Lieb & Loss (section 1.9 of the second edition): It $f_n$ are non-negative and converge a.e. to $f$, then $$ \liminf_n\int f_n = \int f +\liminf_n\int|f-f_n| $$ provided $\sup_n\int f_n<\infty$.