A syntactic approach to Borel functions: Some extensions of Louveau's theorem

Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class $Γ$, then its $Γ$-code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau's theorem to Borel functions: If a Borel function on a Polish space happens to be a $Σ_t$-function, then one can effectively find its $Σ_t$-code hyperarithmetically relative to its Borel code. More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau's theorem for Borel functions.

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