The aim of this work is to study weighted Stepanov-like pseudo almost automorphic functions with infinite delay using the measure theory. We present a new concept of weighted ergodic functions which is more general than the classical one. Then, we establish many interesting results on the space of such functions. We also study the existence and uniqueness of weighted Stepanov-like pseudo almost automorphic solutions of infinite class for some neutral partial functional differential equations in a Banach space when the delay is distributed using the spectral
decomposition of the phase space developed by Adimy and his co-authors. Here, we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition. The delayed part is assumed to be pseudo almost automorphic with respect to the first argument and Lipschitz continuous with respect to the second argument.