On completeness of the space of weighted pseudo almost automorphic functions

In this paper, we prove that for every $\rho\in \mathbb{U}_{\infty}$, the space of weighted pseudo almost automorphic functions is complete under the supremum norm. This gives an affirmative answer to a key and fundamental problem for weighted pseudo almost automorphic functions, and fills a gap in the proof of [J. Funct. Anal. 258, No. 1, 196-207 (2010)].