The aim of this work is to study weighted Stepanov-like pseudo almost periodic 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 functional space of such functions. We also study the existence and uniqueness of Weighted Stepanov-like pseudo almost periodic solutions of infinite class for some neutral partial functional differential equations in a Banach space with infinite delay.