The Kuratowski Extension Theorem says: Let $(X,\mathcal{A})$ be a measurable space, $Y$ be a polish space, $A\subseteq X$, and $f:A\to Y$ be a measurable map. Then there is a measurable function $F:X\to Y$ such that $F|_A=f$.
A proof of the previous theorem can be found at the page 95 of the book "A Course on Borel Sets" written by SM Srivastava.
My quesiton is: Is the previous theorem also true if $Y$ is a Souslin space (i.e., Hausdorff space that is a continuous image of a Polish space)?
I'm asking this question because many theorems of that book can be generalized to Souslin spaces if we use the approach given in chapter 6 of the book "Measure Theory" written by VI Bogachev. However, I didn't find any strategy to generalize that theorem.
