Embedding function spaces into ℓ∞/c0

We show that if the space ℓ∞/c0 contains an isometric copy of every function space over a first countable compactum or every function space over a Corson compactum of weight not exceeding the continuum then every subset of R2 belongs to the σ-field generated by sets of the form A1×A2. We prove a similar result about isomorphic rather than isometric embeddings into ℓ∞/c0 in terms of the σ-field of subsets of Rk generated by sets of the form A1×⋯×Ak for other positive integers k.