  Motivated to generalize the Fourier frame concept to Banach spaces we
introduce (p, q)-Bessel/frame measures for a given finite measure on LCA
groups. We also present a general way of constructing (p, q)-Bessel/frame
measures for a given measure. Moreover, we prove that if a measure has an
associated (p, q)-frame measure, then it must have a certain uniformity in the
sense that the weight is distributed quite uniformly on its support. Next, we
show that if the measures $\mu$ and $\lambda$ without atoms whose supports form
a packing pair, then $\mu\ast\lambda+\delta_g\ast\mu$ does not admit any (p,
q)-frame measure. Finally, we analyze the stability of (p, q)-frame measures
under small perturbations. We prove new theorems concerning the stability of
(p, q)-frame measures under perturbation in both Hilbert spaces and Banach
spaces.
