0
$\begingroup$

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

$\endgroup$
2
  • $\begingroup$ If the sequence of function{fn} is converges uniformly to f the we can exchange.. $\endgroup$
    – Ripan Saha
    Commented Aug 31, 2014 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
    Commented Aug 31, 2014 at 5:04

1 Answer 1

0
$\begingroup$

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.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .