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.
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 $$\sum_{k = 1}^N \inf_{\alpha \ge \overline\alpha_0} x_{k, \alpha} \le \inf_{\alpha \ge \overline\alpha_0} \sum_{k = 1}^\infty x_{k, \alpha} \le \sup_{\overline\alpha} \inf_{\alpha \ge \overline\alpha} \sum_{k = 1}^\infty x_{k, \alpha} = S.$$
