With what extra condition interchange limit and integration allowed?

Question

$f_n$ is sequence of continuous function which converges pointwise to continuous function. with what condition we can interchange limit and integration?

I know that if sequence converges uniformly then we can.But I wanted some what weaker condition .

Any Help will be appreciated.

Answer 1

You might be looking for the dominated convergence theorem, which states that the interchange is permitted if there is an integrable function $g$ such that $|f_n(x)| \leq g(x)$ for all $n \in \mathbb{N}$.