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.