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 Wadgeclass Gamma, then its Gamma-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 Polishspace happens to be a Sigma(t)(sic)-function, then one can find its Sigma(t)(sic)-code hyperarithmeticallyrelative to its Borel code. More generally, we prove extension-type, domination-type,and decomposition-type variants of Louveau's theorem for Borel functions.

Automated summary

Louveau showed that the Gamma-code of a Borel set in a Polish space can be obtained from its Borel code in a hyperarithmetical manner, if it belongs to a Borel Wadge class Gamma. We extend this theorem to Borel functions by proving that the Sigma(t)-code of a Borel function on a Polish space can also be found hyperarithmetically relative to its Borel code. Furthermore, we establish extension-type, domination-type, and decomposition-type variants of Louveau's theorem for Borel functions.