I'm wondering if anyone has shown Fatou's lemma for sums when the limits are taken over nets.


That is, has anyone proved the following?



> 
> Let $\{x\_{k,\alpha}\}\_{k\in \mathbb N,\alpha \in \mathcal A}\subseteq \mathbb R\_+$ with
> $\mathcal A$ a directed set. Then
> 
> 
> $$\sum\_{k=1}^{\infty} \sup\_{\bar \alpha}\inf\_{\alpha\geq \bar \alpha}
>  x\_{k,\alpha}\leq \sup\_{\bar \alpha}\inf\_{\alpha\geq \bar \alpha}
> \sum\_{k=1}^{\infty} x\_{k,\alpha}.$$
> 
> 
> 


This result is not true if the sum were replaced by a general measure.



Fatou's lemma and dominated convergence for nets and the counting measure
If $S \mathrel{:=} \sup\_{\overline\alpha} \inf\_{\alpha \ge \overline\alpha} \sum\_{k = 1}^\infty x\_{k, \alpha}$ were strictly less than $\sum\_{k = 1}^\infty \sup\_{\overline\alpha} \inf\_{\alpha \ge \overline\alpha} x\_{k, \alpha}$, then there would be some $N$ such that $S$ was strictly less than $\sum\_{k = 1}^N \sup\_{\overline\alpha} \inf\_{\alpha \ge \overline\alpha} x\_{k, \alpha}$, and then some $\overline\alpha\_0$ such that $S$ was strictly less than
$$\sum\_{k = 1}^N \inf\_{\alpha \ge \overline\alpha\_0} x\_{k, \alpha} \le \inf\_{\alpha \ge \overline\alpha\_0} \sum\_{k = 1}^\infty x\_{k, \alpha} \le \sup\_{\overline\alpha} \inf\_{\alpha \ge \overline\alpha} \sum\_{k = 1}^\infty x\_{k, \alpha} = S.$$


