Lifting generic points

Let (X, T) and (Y, S) be two topological dynamical systems, where (X, T) has the weak specification property. Let ξ be an invariant measure on the product system (X × Y, T × S) with marginals μ on X and ν on Y, with μ ergodic. Let y ∈ Y be quasi-generic for ν. Then there exists a point x ∈ X generic for μ such that the pair (x, y) is quasi-generic for ξ. This is a generalization of a similar theorem by T. Kamae, in which (X, T) and (Y, S) are full shifts on finite alphabets.