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We study closed subspaces of $L^2(X)$, where $(X, \mu)$ is a $\sigma$-finitemeasure space, that are invariant under the unitary representation associatedto a measurable action of a discrete countable LCA group $\Gamma$ on $X$. Weprovide a complete description for these spaces in terms of range functions anda suitable generalized Zak transform. As an application of our main result, weprove a characterization of frames and Riesz sequences in $L^2(X)$ generated bythe action of the unitary representation under consideration on a countable setof functions in $L^2(X)$. Finally, closed subspaces of $L^2(G)$, for $G$ beingan LCA group, that are invariant under translations by elements on a closedsubgroup $\Gamma$ of $G$ are studied and characterized.

The Zak transform and the structure of spaces invariant by the action of an LCA group