When can I say $\int\limits_{-T}^{T} \sum_n\sum_m f(m.n.t)dt=\sum_n\sum_m \int ^T_{-T}f(m,n,t)dt$?
All I know is that for each $t\in[-T,T]$, $\sum_n\sum_m f(m,n,t)$ is absolutely convergent, which I think is farily strong condition. But now I can't find any easy argument. Could you give me any tips?
I like to remember this as a special case of the Fubini/Tonelli theorems, where the measures are counting measure on $\mathbb{N}$ and Lebesgue measure on $\mathbb{R}$ (or $[0,\infty)$ as you've written it here). In particular, Tonelli's theorem says if $f_n(x) \ge 0$ for all $n,x$, then $$\sum \int f_n(x) \,dx = \int \sum f_n(x) \,dx$$ without any further conditions needed. (You can also prove this with the monotone convergence theorem.)
Then Fubini's theorem says that for general $f_n$, if $\int \sum |f_n| < \infty$ or $\sum \int |f_n| < \infty$ (by Tonelli the two conditions are equivalent), then $\int \sum f_n = \sum \int f_n$. (You can also prove this with the dominated convergence theorem.)
