Motivated by the recent works by the first and the second named authors, in this paper we introduce the notion of doubly-weighted pseudo-almost periodicity (respectively, doubly-weighted pseudo-almost automorphy) using theoretical measure theory. Basic properties of these new spaces are studied. To illustrate our work, we study, under Acquistapace–Terreni conditions and exponential dichotomy, the existence of (μ,ν)- pseudo almost periodic (respectively, (μ,ν)-pseudo almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces. A few illustrative examples will be discussed at the end of the paper.