Suppose $f_n$ converges pointwise to $f$ as $n\rightarrow\infty$, and that $\int{f}=\infty$.
Must it be the case that there is some $N$ such that for $n>N$, $\int{f_n}>0$? Or at least that $\int{f_n}>0$ for infinitely many $n$?
Note that since $\int{f}=\infty$ it appears the dominated convergence theorem does not apply directly (even if we had the requisite "$g$" function).
While I have a particular (incredibly messy) $f_n$ in mind, I'd certainly be interested in counter-examples if this isn't true.
Trivial counterexample of a function which is $1$ on $[0,n)$ and $-1$ on $[n,2n+1)$, $0$ everywhere else. Integral always $-1$. Pointwise limit is $1$ on $[0,\infty)$.