I'm pretty bad at math, so I need some help reasoning through this. I'm trying to do a proof, and I think I used some flavor of the below ideas in my proof. I decided in the meantime to find another path for my proof because I'm questioning whether the ideas hold any water.
Suppose $f$ and $(f_n)$ are non-negative bounded $\mathcal F$-measurable and we know that $\forall F\in\mathcal F$, $$\int_Ff\,\mathrm d\mu\leq\liminf\int_Ff_n\,\mathrm d\mu$$ I wonder if this implies $f\leq\liminf f_n$ almost everywhere.
On a similar note, if $\forall F\in\mathcal F$, $$\lim\int_Ff_n\,\mathrm d\mu=\int_Ff\,\mathrm d\mu$$ then $f_n\to f$ almost everywhere.
The reason why I ask is because the $\forall$ quantifier seems very strong, and suggests that the desired relations hold because behavior under Lebesgue integrals is very "steady" and holds no matter what region I look at, i.e., that if it's not true, then it's on negligible sets.
On the other hand, these remind me of converses of Fatou's lemma and the dominated convergence theorem. So I'm also suspicious.