Conditions for interchanging order of limits and summations Let $f: \mathbb{N} \times \mathbb{N} \to \overline{\mathbb{R}}$. Then under which conditions is the expression $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$ valid?
Would anyone have a rigorous answer to this? Any proof using measure theory, or elementary calculus, is more than welcome.
I know that a very similar question has been asked here: Under what condition we can interchange order of a limit and a summation? , but I would need more detail. For example, one of the answers states that the dominated convergence theorem suffices as 'sums are just integrals with respect to the counting measure on $\mathbb{N}$'. I am unable to see how works; I don't know how this 'counting measure' can be used with the dominated convergence theorem to provide the conclusion.
 A: While looking for higher considerations let me suggest some simple criteria.

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*Suppose we have double sequense $a_{n,m}$. If exists $\lim_\limits{m \to \infty}a_{n,m}=a_n, \ n\in \mathbb{N}$ ; and series $\sum\limits_{n=1}^{\infty}a_{n,m}$ converged uniformly, then we can interchange order of limit and summation.


*Suppose series $\sum\limits_{m,n =1}^{\infty}a_{n,m}$, $\sum\limits_{n =1}^{\infty}\sum\limits_{m =1}^{\infty}a_{n,m}$ and $\sum\limits_{m =1}^{\infty}\sum\limits_{n =1}^{\infty}a_{n,m}$ all converged. Then they equal one and same value.


*Suppose $f(x,y)$ is defined on some set $E$, which includes all points from some rectangle with center in $(x_0,y_0)$, except, possibly, lines $y=y_0$ and $x=x_0$. If exists double limit for $f$ with respect to $E$ and for any $y \ne y_0$ in some neigbourhood of $y_0$ exists $\lim\limits_{x \to x_0}f(x,y) = g(y)$, then exists $\lim\limits_{y \to y_0}g(y)$ and holds
$$\lim\limits_{y \to y_0}\lim\limits_{x \to x_0}f(x,y) = \lim\limits_{(x,y) \to (x_0,y_0)}f(x,y)$$
