On monotone presentations of Borel sets

If A is a Σξ set and An (n < ω) are Borel sets then we call {An : n < ω} a presentation of A if A = S n<ω An and An (n < ω) have lower Borel class than A has. We show that for 2 ≤ ξ < ω1 it is not possible to assign a presentation to Σξ sets in a monotone way, i.e., it is not possible to define functions fn : Σ 0 ξ → Πξ (n < ω) such that for every A ∈ Σξ we have A = S n<ω fn(A) and A,A ′ ∈ Σξ, A ⊆ A′ implies fn(A) ⊆ fn(A) (n < ω). This answers a question of Márton Elekes in the negative. We also show the nonexistence of monotone presentation for Borel functions.