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A simple question: when can wo exchange sum and integral? $$\sum_{n=0}^\infty\int f_n(x)dx=\int\sum_{n=0}^\infty f_n(x)dx=\int f(x)dx$$ $$\lim_{n\rightarrow \infty}\int f_n(x)dx=\int\lim_{n\rightarrow \infty} f_n(x)dx=\int f(x)dx$$

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  • $\begingroup$ If the sequence of function{fn} is converges uniformly to f the we can exchange.. $\endgroup$ – Ripan Saha Aug 31 '14 at 4:32
  • $\begingroup$ Since the other posters hit the big condition, I'll add: in the context of Riemann integration, there is a weakened version of Lebesgue's dominated convergence theorem, math.washington.edu/~morrow/335_14/dominated.pdf. Further, while this is not theoretically that useful in my experience, pointwise convergence is sufficient if it is known that the limit function is Riemann integrable. $\endgroup$ – GWilliams Aug 31 '14 at 5:04
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Exchange sum and integral: For Riemann's integral, it is necessary that the serie $\sum f_n$ converge uniformly. For Lebesgue's integral, it suffices that each function $f_n$ is positive.

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