Double Limit Swapping With Both Sum and Integral For now, say that we have a sequence of real-valued $L^2$ functions $(f_n)$, defined on $\mathbb{R},$ converging to $f\in L^2$ pointwise and the sequence is pointwise bounded by an $L^2$ function. Consider the following limit:
$$\lim_{n\rightarrow\infty}\sum\limits_{m=0}^\infty\left|\left(\int_{K_m} |f_n(x)|^2\, dx\right)^{1/2}\right|,$$ where $\{K_m\}$ is just some collection of compact sets with union $\mathbb{R}$. Under what conditions can we move the limit inside both the sum and the integral? I'm sure it's intimately related to dominated convergence, but I'd appreciate it if someone could write it explicitly for me. 
I currently have it set up so that I can swap the limit and the integral, so I just have to justify swapping the limit and the sum. Do I just do something like define a sequence $$g_m(x)=\left(\int_{K_m} |f_n(x)|^2\, dx\right)^{1/2},$$ then require that it converges pointwise and is bounded a.e. by an $\ell^1$ function? 
The conditions on $(f_n)$ and $f$ are pretty loose, so we can take them to be fairly nice, if needed (Schwartz, for example).
 A: The interchange need not be justified. Consider domain $(1,\infty) \subseteq \mathbb{R}$, $K_m = [m,m+1], f_n(x) = \frac{1}{nx}, f = 0$. Then $f_n \in L^2$ for each $n$, $f \in L^2$, $f_n \to f$ pointwise, and the $f_n$'s are pointwise bounded by $\frac{1}{x} \in L^2$. Then $\int_{K_m} |f_n(x)|^2dx \approx \frac{1}{n^2m^2}$, so $\sum_{m=1}^\infty (\int_{K_m} |f_n(x)|^2dx)^{1/2} \approx \sum_{m=1}^\infty \frac{1}{nm} = +\infty$, so $\lim_n \sum_m \cdot = +\infty$, whereas $\sum_m \lim_n \cdot \approx \sum_m \lim_n  \frac{1}{nm} = \sum_m 0 = 0$.
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Remark: If we instead had $\lim_n \sum_m \int_{K_m} |f_n(x)|^2dx$, then we could interchange the limit and the sum. Indeed, if each $f_n$ is pointwise dominated by some $g \in L^2$, then for each $n$, $\int_{K_m} |f_n(x)|^2dx \le \int_{K_m} |g(x)|^2 dx$, so, for each $n$, the function $m \mapsto \int_{K_m} |f_n(x)|^2dx$ is bounded by $\int_{K_m} |g(x)|^2dx$, which is an $L^1$ function in the relevant sense: $\sum_m \int_{K_m} |g(x)|^2dx = \int_{\mathbb{R}} |g(x)|^2dx < \infty$, so dominated convergence theorem allows one to interchange $\lim_n$ and $\int_m = \sum_m$.
