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Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with finite integration is possible?

$$\int_1^B\sum_{1}^{N} f_n(t)dt = \sum_{1}^{N} \int_1^B f_n(t)dt .$$

Question (2) After we take the limits of on both sides of the equation above, do we get the same limits?

$$\lim_{B\rightarrow\infty}\lim_{N\rightarrow\infty}\int_1^B\sum_{1}^{N} f_n(t)dt = \lim_{N\rightarrow\infty}\lim_{B\rightarrow\infty}\sum_{1}^{N} \int_1^B f_n(t)dt .$$

Thanks- Mike

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  • $\begingroup$ Why $(t,s)$? Is one the variable of integration and the other a parameter? But why mention the parameter? $\endgroup$
    – Robert Israel
    Commented May 1, 2014 at 6:54
  • $\begingroup$ @Robert: Thanks a lot for you comment. I deleted the parameter $s$ in the questions. $\endgroup$
    – mike
    Commented May 1, 2014 at 13:38

1 Answer 1

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1) See Fubini's Theorem. You want $\sum_1^N \int_1^B |f_n|$ to be finite. This is just the case where one of the two measures is counting measure.

2) Again, Fubini's theorem, this time with $\sum_{1}^\infty \int_1^\infty |f_n|$.

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