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I have a real function, $f(n,m)$, which is not necessarily bounded nor necessarily non-negative, but has point-wise convergence of: \begin{equation*} g(m) = \sum\limits_{n=1}^{\infty} f(n,m) \end{equation*} and $g(m)$ is finite for all integers $m$.

I am examining the interchange of limits in \begin{equation*} \sum\limits_{m=1}^{\infty} \sum\limits_{n=1}^{\infty} f(n,m) \sim \sum\limits_{n=1}^{\infty} \sum\limits_{m=1}^{\infty} f(n,m) \end{equation*} and will settle for bounding the summation within a (possibly infinite) range.

If I define: \begin{equation*} g_{N}(m) = \inf_{K \ge N} \sum\limits_{k=1}^{K} f(k,m) \le g(m) \end{equation*} and \begin{equation*} h_{N}(m) = \sup_{K \ge N} \sum\limits_{k=1}^{K} f(k,m) \ge g(m) \end{equation*}

Then, can I state the following?

IF $\sum\limits_{m=1}^{\infty} g(m)$ converges, then the limit is within the (possibly infinite) range of: \begin{equation*} \begin{aligned} \liminf_{N \to \infty} \liminf_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{N} f(n,m) &\le \lim_{M \to \infty} \sum\limits_{m=1}^{M} g(m) \\ &\le \limsup_{N \to \infty} \limsup_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{N=1}^{N} f(n,m) \end{aligned} \end{equation*}

My reasoning is thus: \begin{equation*} \begin{aligned} \liminf_{N \to \infty} \liminf_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{N} f(n,m) &\le \liminf_{M \to \infty} \liminf_{N \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{N} f(n,m) \\ &\le \liminf_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{N=1}^{\infty} g_{N}(m) \\ &\le \liminf_{M \to \infty} \sum\limits_{m=1}^{M} g(m) \\ &\le \lim_{M \to \infty} \sum\limits_{m=1}^{M} g(m) \\ &\le \limsup_{M \to \infty} \sum\limits_{m=1}^{M} g(m) \\ &\le \limsup_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{N=1}^{\infty} h_{N}(m) \\ &\le \limsup_{M \to \infty} \limsup_{N \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{N=1}^{N} f(n,m) \\ &\le \limsup_{N \to \infty} \limsup_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{N=1}^{N} f(n,m) \end{aligned} \end{equation*}

With the understanding that the $\liminf$ or $\limsup$ operations could diverge to $\pm \infty$ or converge to different values, the statement above is similar to Fatou's Lemmas, but with the counting measure and with a much, much weaker convergence statement.

My question:

Is this weak, weak convergence statement true for all such $f(n,m)$? If it is not true, can a counter example be provided?

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Herea a counterexample:

Let $f(n,m)$ be defined as follows. For fixed $m$, the sequence $f(n,m)$ is

$$f(n,m)=\left\{\begin{array}[l]\ 1\ \ \ \text{for}\ \ \ n\leq m\\ -m\ \ \ \text{for }\ \ \ n=m+1\\ 0\ \ \ \text{for}\ \ \ n>m+1\end{array}\right.$$

Thus $g(m)=0$ for any $m$ and hence $\sum g(m)=0$.

On the other hand for any $N$,

$$\liminf_{M\to\infty} \sum_{m=1}^M\sum_{n=1}^N f(n,m)=\infty$$ because

$$\sum_{m=1}^M\sum_{n=1}^Nf(n,m)=\sum_{m\leq N}\sum_{n=1}^Nf(n,m)+\sum_{m>N}\sum_{n=1}^Nf(n,m)=0+(M-N)\sum_{n=1}^N 1=(M-N)N$$

therefore $$\liminf_{N\to\infty}\liminf_{M\to\infty} \sum_{m=1}^M\sum_{n=1}^N f(n,m)=\liminf_{N\to\infty}\infty=\infty$$

and therefore the claim $$\liminf_{N\to\infty}\liminf_{M\to\infty} \sum_{m=1}^M\sum_{n=1}^N f(n,m)\leq\sum g(m)$$ is false

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  • $\begingroup$ With the provision of a counter example the posted question is answered in the negative. NO, such a point-wise convergent series cannot be bounded by \liminf and limsup unless the function is non-negative. $\endgroup$ May 29, 2014 at 15:57
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Consider this infinite table: $$ \begin{pmatrix} * & 0 & 0 & 0 & 0 & 0 & \dots \\ * & * & 0 & 0 & 0 & 0 & \dots \\ 0 & * & * & 0 & 0 & 0 & \dots \\ 0 & 0 & * & * & 0 & 0 & \dots \\ 0 & 0 & 0 & * & * & 0 & \dots \\ 0 & 0 & 0 & 0 & * & * & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix} $$ If you put numbers for the stars you can construct all systems of row and column sums. The sequences $R_n=\sum\limits_{k=1}^\infty f(n,k)$ and $C_n=\sum\limits_{n=1}^\infty f(n,k)$ can be any sequences, independently.

So, without additional assumptions there is no such statement on changing the order of summations.

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  • $\begingroup$ To User user141614: with your counter example, this question was worth every one of the 100 reputation points I expended. These are great counter examples and your with the Matrix exposition is crystal clear that without additional assumptions the row and column sequences have no relationship to each other and there is no guarantee that: \sum_{n=1}^{\infty} R_n = \sum_{k=1}^{\infty} C_k $\endgroup$ Aug 8, 2014 at 21:35

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