Sequences and Series/Multiple limits

interchanging summation and integration Let be a measure space, and let be a sequence of functions from to , where or . If either of the two expressions or converges, so does the other, and we have . Regarding the summation as integration over with σ-algebra and counting measure, this theorem is an immediate consequence of Fubini's theorem, given that integration and summation are defined pointwise. interchanging summation and real differentiation Let be a sequence of continuously differentiable functions from an open subset of to . Suppose that both and converge for all , and that for all there exists and a sequence in such that and . Then for all .