Theorem (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
converges, so does the other, and we have
Proof: 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.
Theorem (interchanging summation and real differentiation):
Let be a sequence of continuously differentiable functions from an open subset of to . Suppose that both
converge for all , and that for all there exists and a sequence in such that
- and .
for all .