Statement of the DCT: Let
tend pointwise to
, with each
absolutely integrable. If we also have
with
(absolutely) integrable,
.
That could be almost any kind of integral, including infinite sums. It's also closely related to Fubini's
theorem. In the usual Lebesgue context, we relax "pointwise" to "pointwise almost everywhere". For improper Riemann integrals, that's "pointwise except on a discrete set".
The idea of the proof is to look at
, and split this into two parts. On one part, we make it small with the pointwise
convergence. On the other, we make it small by comparing to
and making the region small.