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.
Best Answer
While looking for higher considerations let me suggest some simple criteria.
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 $$\lim\limits_{y \to y_0}\lim\limits_{x \to x_0}f(x,y) = \lim\limits_{(x,y) \to (x_0,y_0)}f(x,y)$$