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.