A trichotomy for a class of equivalence relations

Abstract

Let X _n , n ∈ ℤ be a sequence of non-empty sets, ψ _n : X ² _n → ℝ⁺. We consider the relation E = E((X _n , ψ _n ) _n∈ℤ) on Π _n∈ℤ X _n by (x, y) ∈ E((X _n , ψ _n ) _n∈ℤ) ⇔ Σ _n∈ℤ ψ _n (x(n), y(n)) < + ∞. If E is an equivalence relation and all ψ _n , n ∈ ℤ, are Borel, we show a trichotomy that either ℝ^ℤ/ℓ ₁ ⩽ _B E, E ₁ ⩾ _B E, or E ⩾ _B E ₀.

We also prove that, for a rather general case, E((X _n , ψ _n ) _n∈ℤ) is an equivalence relation iff it is an ℓ _p -like equivalence relation.