Modulation invariant spaces on locally compact abelian groups

We define and investigate modulation invariant spaces on a locally compact abelian group $G$ with respect to a closed subgroup of the dual group $\widehat{G}$. Using a range function approach, we establish a characterization of modulation invariant spaces. Finally, we define a metric on the collection of all modulation invariant spaces and study some topological properties of the metric space.