The interchange of limit for integration This kind of problem bothers me for a while. Each time I meet such problem I got stuck and has to deal them case by case. So I post this problem here to ask for some general condition of the "interchangeable" of limit.
Let $f_n\to f$ and $g_m\to g$ in $L^2$ be given, then in what condition that I can do
$$\lim_{n\to\infty}\lim_{m\to\infty}\int f_n g_m dx= \lim_{m\to\infty}\lim_{n\to\infty}\int f_n g_m dx$$
Or in general, what condition that $F_{n,m}$ has to satisfy so that 
$$\lim_{n\to\infty}\lim_{m\to\infty}\int F_{n,m} dx= \lim_{m\to\infty}\lim_{n\to\infty}\int F_{n,m} dx$$
Any reference or help would be very welcome!
 A: To elaborate on the comment by @Pietro Majer:
The Cauchy-Schwarz inequality yields
\begin{eqnarray*}
\left|\int f_{n}g_{m}-\int fg\right| & \leq & \left|\int\left(f_{n}-f\right)g_{m}\right|+\left|\int f\left(g_{m}-g\right)\right|\\
 & \leq & \left\Vert f_{n}-f\right\Vert _{2}\cdot\left\Vert g_{m}\right\Vert _{2}+\left\Vert f\right\Vert _{2}\cdot\left\Vert g_{m}-g\right\Vert _{2}\\
 & \leq & C\cdot\left\Vert f_{n}-f\right\Vert _{2}+\left\Vert f\right\Vert _{2}\cdot\left\Vert g_{m}-g\right\Vert _{2}\xrightarrow[n,m\rightarrow\infty]{}0,
\end{eqnarray*}
where the last line used that a convergent sequence is bounded.
Note that the convergence holds regardless of the exact order in which you take the limits (first $n \to \infty$, then $m \to \infty$ or vice versa or $(n,m) \to \infty$ "simultaneously").
EDIT: I think the second question is too general to admit a satisfactory answer. What kind of conditions do you have in mind?
For example, it would suffice if $\lim_{(n,m) \to \infty} F_{n,m}(x) = f(x)$ for some $f$ and also $|F_{n,m}(x)| \leq g(x)$ for some integrable $g$ (essentially, this is dominated convergence, the exact question was posed a while ago, here: Dominated convergence for sequences with two parameters, i.e. of the form $f_{m,n}$).
