In that case, what do you mean by a.e. convergence? If it means almost everywhere with respect to the Borel measure, then surely the usual proof that a (pointwise a.e.) limit of a sequence of measurable functions is measurable will work for that measure? On the other hand, if it means a.e. (Lebesgue) convergence then I think you can have a rather trivial example, like this. Let f be the characteristic function of a Borel non-measurable subset B of the Cantor set, and let f_n be the zero function, for all n. Then f_n → f a.e. (because f is also zero a.e.!), each f_n is Borel measurable, but f is not.