Fatou's lemma says that
$$ \liminf_{n\to\infty}\int_0^tf_n(s)ds\ge \int_0^t\liminf_{n\to\infty}f_n(s)ds $$
provided that there exists a nonnegative integrable function $g$ such that $f_n\ge-g$ for all $n$. In addition, if $f_n$ is, for each $n$, lower semicontinuous function, then
$$ \liminf_{n\to\infty}\int_0^tf_n(s)ds\ge \int_0^t f(s)ds, $$ where $t>0$ and $\liminf_{n\to\infty}f_n\ge f$.
I know the above statement. However, as soon as $t$ depend on the index, i.e., $t_n$, I don't understand. More precisely, the problem is whether
$$ \liminf_{n\to\infty}\int_0^{t_n}f_n(s)ds\ge\int_0^tf(s)ds $$
is true or not. Here, $\lim t_n=t$.
I tried to apply the above way for $g_n(s):=f_n(s)1_{[0,t_n]}(s)$ but, originally it is not trivial that $\liminf_{n\to\infty}g_n\ge f(s)1_{[0,t]}(s)$ since $f_n$ may be negative. Here $1_{I}$ is the indicator function on $I$.d
I believe that the above inequality holds. Please any advices, comments or solutions. Thank you in advance.