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As we know, uniform convergence on a finite interval can lead to the interchange of infinite sum with integration. However, it seems that for improper integration, stronger conditions are needed. For example, if $f_n(x)$ uniformly converges to $f(x)$, such that $|f_n(x)-f(x)| <g(x)\epsilon$ for all $n>N$, where $\int g(x)$ is convergent, then $\int|f_n(x)-f(x)|<C\epsilon$ for any $n>N$. In such case, interchange the limit (in the case of series, it's interchanging summation) with improper integration is available.

Is there any theorem about this that is user friendly?

ZWJ
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  • https://math.stackexchange.com/questions/83721/when-can-a-sum-and-integral-be-interchanged –  Dec 30 '20 at 16:38
  • https://math.stackexchange.com/questions/1334907/reversing-the-order-of-integration-and-summation –  Dec 30 '20 at 16:38
  • does any of the above two links answer your question? Both are about interchanging sum with indefinite integral –  Dec 30 '20 at 16:39
  • As written the question is too broad. Are you concerned with improper Riemann integrals or more generally Lebesgue integrals? Much is covered by the dominated and monotone convergence theorems, but there can be special cases where these are not applicable. What is a "user friendly" theorem and why would you expect one to exist that covers all possibilities? – RRL Dec 30 '20 at 18:14
  • @RRL Improper Riemann integral...As for 'user friendly', I mean for example, some sufficient conditions on $f_n(x)$ (that are easy to check) that can make the interchange available... – ZWJ Dec 31 '20 at 04:26

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