VITALI'S THEOREM WITHOUT UNIFORM BOUNDEDNESS

Let {f(m)}m >= 1 be a sequence of holomorphic functions defined on a bounded domain D subset of C-n or a sequence of rational functions (1 <= deg r(m) <= m) defined on C-n. We are interested in finding sufficient conditions to ensure the convergence of {f(m)}m >= 1 on a large set provided the convergence holds pointwise on a not too small set. This type of result is inspired from a theorem of Vitali which gives a positive answer for uniformly bounded sequence.