The aim of this work is to present new concept of square-mean pseudo almost automorphic of infinite class using
the measure theory. We use the (μ, ν)-ergodic process to define the spaces of (μ, ν)-pseudo almost automorphic processes of infinite class in the square-mean sense. We present many interesting results on those spaces like completeness and composition theorems and we study the existence and the uniqueness of the square-mean (μ, ν)-pseudo almost automorphic solutions of infinite class for of the stochastic evolution equation. We provide an example to illustrate ours results.