Given that something holds for an arbitrary union of sets, does it also hold for the lim sup?

In a problem I am working on, I want to say the following:

Since $int f chi_{[0, 1] setminus cup_{i = l}^{infty}I_{n_i}} = 0$ for each $l$, $int f chi_{[0, 1] setminus cap_{l = 1}^{infty}cup_{i = l}^{infty}I_{n_i}} = 0$

Is this true? Note that the $I_{n_i}$ are all disjoint open sub-intervals of $[0, 1]$.