Duality and stable compactness in Orlicz-type modules

Orlicz-type modules are module analogues of classical Orlicz spaces. We study duality and stable compactness in Orlicz-type modules. We characterize the conditional K"{o}the dual of an Orlicz-type module as the space of all $\sigma$-order continuous module homomorphisms. We find an order continuity criterion for stable compactness in Orlicz-type modules. As an application, we obtain a robust representation result for conditional risk measures on Orlicz spaces.