  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., $|{\rm RM}|\sim10^5~{\rm rad~m^{-2}}$. 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 non-relativistic electrons that contribute to RM. The non-detections
of persistent emission sources from all other localized FRB sources are
consistent with their relatively small RMs ($\left|{\rm RM}\right|\lesssim{\rm
a~few}\times100~{\rm rad~m^{-2}}$) 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.
