Let $f_n : [0,1] \to \mathbb{R}$ be a sequence of integrable functions such that \begin{equation} \int_{t_1}^{t_2} f_n(t) dt \to a_{t_1, t_2} \text{ as } n \to \infty \end{equation} for a.e. $t_1 \in [0,1]$ and all $t_2 \in [0,1]$ such that $t_2 \geq t_1$. Here, $a_{t_1, t_2} \in [0,\infty)$ for all such $t_1, t_2$'s.
Then, I wonder if the limit still exists for any "open subset" $U \subset (0,1)$.
This seems quite confusing since there is no nonnegativity assumed about $f_n(t)$'s even though $U$ may be expressed as a countable union of disjoint open intervals..
Could anyone please help me?