We revisit the Banach space \(PAP(\mathbb {R},X,\mu ,\nu )\) of pseudo almost periodic functions with respect to two measures \(\mu \) and \(\nu \) introduced in [22]. Using the Banach–Steinhaus theorem, we give a detailed study of the relationship between the ergodic spaces \(\mathcal {E}(\mathbb {R},X,\mu )\) and \(\mathcal {E}(\mathbb {R},X,\mu ,\nu )\) . Moreover, we correct a claim of the uniqueness of the decomposition \(PAP(\mathbb {R},X,\mu ,\nu )=AP(\mathbb {R},X)\oplus \mathcal {E}(\mathbb {R},X,\mu ,\nu )\) into almost periodic and ergodic parts by showing that the asymptotic joint growth condition \({\displaystyle \underset{r\rightarrow \infty }{\limsup }}\frac{\mu ([-r,r])}{\nu ([-r,r])}>0\) is necessary and sufficient for this unique decomposition to hold. Furthermore, we provide a sufficient condition on the nonlinearity which ensures the existence and uniqueness of a solution within this class of functions for certain evolution equations. This result is proved using Bielecki-type weighted norms.
