Dominated convergence theorem

In measure theory, Lebesgue's dominated convergence theorem gives a mild sufficient condition under which limits and integrals of a sequence of functions can be interchanged. More technically it says that if a sequence of functions is bounded in absolute value by an integrable function and is almost everywhere pointwise convergent to a function then the sequence converges in L 1 {\displaystyle L_{1}} to its pointwise limit, and in particular the integral of the limit is the limit of the integrals.

Source: Wikipedia — Dominated convergence theorem (CC BY-SA 4.0)