In my particular case, suppose that $X$ is a curve smooth (and other good adjectives.) over a finite field $\mathbb F_q$ and $G$ be a divisor.
The space $L_k(G)=\{\mathbb F_q (X):(f)+G\geq 0\}$ is a sub-space of $L(G)=\{\overline{\mathbb F_q}(X):(f)+G\geq 0\}$.
$L_k(G)$ is a finite set right?
Or we should add some hypothesis in G?