I am trying to understand the definition of a Gabor frame and would appreciate some clarification with terminology. Let us begin with the setup: Let $G$ be a locally compact abelian group, and let $\widehat{G}$ be its Pontryganin dual. Let $\mu$ be a (left) Haar measure, denote the associated square integrable functions by $L^2(G)$ and let $$ \pi:G \times \widehat{G} \to \mathcal{B}(L^2(G)) $$ a continuous group representation. Finally, let $\Delta \subseteq G \times \widehat{G}$ be a closed cocompact subgroup.
For some fixed element $g \in L^2(G)$, we say that $(\pi(z)g)_{z \in \Delta}$ is a continuous Gabor frame is it is weakly measurable and $\exists$ $C,D > 0$ such that $$ C\|f\|^2 \leq \int_{\Delta}|\langle f,\pi(z),g\rangle|^2dz \leq D\|f\|^2, $$ where the integral is interpreted weakly.
What do the highlighted terms weakly measurable and interpreted weakly mean? I know what a weakly measurable function from a measure space to a Banach space is, and I would guess that here weakly measurable means that the function $$ \pi(-)g:\Delta \to \mathcal{B}(L^2(G)) $$ is weakly measurable. Is this correct? As for interpreted weakly I really have no idea what this means . . .
