Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class \(\Gamma \), 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 Polish space happens to be a \( \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t\)-function, then one can find its \( \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t\)-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.

Louveau 表明，如果波兰空间中的 Borel 集恰好位于 Borel Wadge 类\(\Gamma \)中，则可以通过超算术方式从其 Borel 代码中获得其\(\Gamma \)代码。我们将 Louveau 定理扩展到 Borel 函数：如果波兰空间上的 Borel 函数恰好是\( \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t\) -函数，那么一个可以找到其\( \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t\) -相对于其 Borel 代码的超算术代码。更一般地，我们证明了波雷尔函数的 Louveau 定理的扩展型、支配型和分解型变体。