Let $E_{1}, E_{2}, \dots$ be a collection of measurable sets in a finite measure space, with the property that $m(E_{i} \cap E_{j}) = 0$ whenever $i \neq j$. I proved that $$m \left( \bigcup_{j=1}^{\infty} E_{j}\right) = \sum_{j=1}^{\infty} m(E_{j})$$
by taking the limit as $n \to \infty$ of the Inclusion-Exclusion formula. But then I realized that this formula need not hold for countable collections, since the series involved may not converge. But given that all the terms involving intersections are of measure zero, is there anything actually wrong with my argument?
I can prove this result another way with no issue, I just wanted to know if my first approach is at all problematic.