Decomposition of weighted pseudo-almost periodic functions

First, we show by constructing two counterexamples that the decomposition of weighted pseudo-almost periodic functions is not unique in general. Then we prove that the decomposition of such functions is unique if PAP0(X,ρ)PAP0(X,ρ) is translation invariant, but not necessarily unique without the assumption. Moreover, we give an example to show that the mean value under a certain weight ρρ may not exist for all almost periodic functions. With these results, we answer some fundamental questions on weighted pseudo-almost periodic functions.