Effective refining of Borel coverings

Given a countable family ( Γ i ) i ∈ I (\mathbf {\Gamma }_i)_{i\in I} of additive or multiplicative Baire classes ( Γ i = Σ ξ i 0 \mathbf {\Gamma }_i=\mathbf {\Sigma }^0_{\xi _i} or Π ξ i 0 \mathbf {\Pi }^0_{\xi _i} ) we investigate the following complexity problem: Let ( A i ) i ∈ I (A_i)_{i\in I} be a Borel covering of ω ω \omega ^\omega and assume that there exists some covering ( B i ) i ∈ I (B_i)_{i\in I} with B i ⊂ A i B_i\subset A_i and B i ∈ Γ i B_i\in \mathbf {\Gamma }_i for all i i ; can one find such a family ( B i ) i ∈ I (B_i)_{i\in I} in Δ 1 1 ( α ) \varDelta ^1_1(\alpha ) where α ∈ ω ω \alpha \in \omega ^\omega is any reasonable code for the families ( A i ) i ∈ I (A_i)_{i\in I} and ( Γ i ) i ∈ I (\mathbf {\Gamma }_i)_{i\in I} ? The main result of the paper will give a full characterization of those families ( Γ i ) i ∈ I (\mathbf {\Gamma }_i)_{i\in I} for which the answer is positive. For example we will show that this is the case if I I is finite or if all the Baire classes Γ i \mathbf {\Gamma }_i are additive, but in the general case the answer depends on the distribution of the multiplicative Baire classes inside the family ( Γ i ) i ∈ I (\mathbf {\Gamma }_i)_{i\in I} .