In this chapter, we investigate a class of HJB equations defined over infinite horizon. Our analysis is conducted under a set of mild assumptions, which are sufficient to derive a representation theorem tailored to the case of fiber-convex Hamiltonians. This representation result plays a crucial role in the subsequent development, as it enables us to establish a uniqueness property for weak solutions that vanish at infinity. Particular attention is given to Hamiltonians that are time-measurable, which is a typical features in many control models.