The physical origin of fast radio bursts (FRBs) is still unknown. Multiwavelength and polarization observations of an FRB source would be helpful to diagnose its progenitor and environment. So far only the first repeating source FRB 121102 appears to be spatially coincident with a persistent radio emission. Its bursts also have very large values of the Faraday rotation measure (RM), i.e., . We show that theoretically there should be a simple relation between RM and the luminosity of the persistent source of an FRB source if the observed RM mostly arises from the persistent emission region. FRB 121102 follows this relation given that the magnetic field in the persistent emission region is highly ordered and that the number of relativistic electrons powering the persistent emission is comparable to that of nonrelativistic electrons that contribute to RM. The nondetections of persistent emission sources from all other localized FRB sources are consistent with their relatively small RMs ( ) according to this relation. Based on this picture, the majority of FRBs without a large RM are not supposed to be associated with bright persistent sources.