Measure theory was introduced in the early 1900s by Lebesgue, at the same time with Hausdorff introducing the usual concept of topology, and publishing it in his book just before World War I. Measure theory is full with convergence and limit properties of measures, functions and integrals. Yet none of them uses usual toplogy. Is there a well thought out reply to why such a thing happens ?
It is true that convergence a.e. is not the convergence of a topology. If you want a topology you can go to convergence in measure. A theorem MORE GENERAL THAN the dominated convergence theorem: let $f_n \to f$ in measure on a set $E$ of finite measure, and suppose there is an integrable $g$ with $|f_n| \le g$ a.e. Then $\int_E f_n \to \int_E f$. SPECIAL CASE: $f_n \to f$ a.e.