On Projective Mappings

Let $X,Y$ be Polish spaces, and let $\mathscr B_k$ be the $\sigma$-algebra generated by the projective class $L_{2k+1}$. A mapping $f\colon X\mapsto Y$ is called $K$-projective if $f^{-1}(E)\in \mathscr B_k$ for any Borel subset $E\subset Y$. The following theorem is our main result: for any $k$-projective mapping $f\colon X\mapsto Y$ there exist a Polish space $\widetilde X_S$, a dense subset $X_S\in \mathscr B_k$, and two continuous mappings $f_0, i: \widetilde X_S\to Y$ such that

• i) $f_0|_{X_S}=f\circ i|_{X_S}$;
• ii) $i|_{X_S}$ is a bijection.