Extension of Borel measurable function on a subset of a Polish space I bump into the following theorem and get lost at the very first step of its proof.
Theorem: Let $X$ and $Y$ be Polish spaces, $A \subset X$, $B \subset Y$, and let $f:A \rightarrow B$ be a Borel isomorphism, i.e., a one-to-one Borel mapping such that $f^{-1}$ is Borel measurable provided that $A$ and $B$ are equipped with the induced Borel $\sigma$-algebra. Then one can find two sets $A^{\star} \in \mathscr{B}(X)$ and $B^{\star} \in \mathscr{B}(Y)$ and a Borel isomorphism $f^{\star}: A^\star \rightarrow B^{\star}$ such that $A \subset A^\star$, $B \subset B^\star$ and the restriction of $f^\star$ to $A$ is $f$.
At the first step, the proof says "clearly, one can find Borel mappings $f^{\star}: X \rightarrow Y$ such that the restriction of $f^\star$ to $A$ is $f$". I'm lost here. My question is: since $A$ might be any subset (and hence might not be Borel), how to show the existence of such extension $f^\star$?
 A: This is in fact the exercise 8.13 of Donald L. Cohn's measure theory book. And $X$ need not to be a polish space. And the paper THE EXTENSION OF MEASURABLE FUNCTIONS by R. M. SHORTT studied the problem in advance. However, he only gave a fairly rough sketch and cited the exercise of Donald L. Cohn. He wrote functions assuming finitely many values and pass to the limit. I can only provide the proof of a weaker result using this idea, as follow.
It is known that every Polish space is homeomorphic to a $G_\delta$ in $[0,1]^\mathbb{N}$. I can prove that if $Y$ is $[0,1]^\mathbb{N}$, then the result holds. Without loss of generality, we can assume $Y$ is $[0,1]$ and $f^*$ is real value (since in general case one can work with each dimension separately). We can also assume $f^*$ is nonnegative. Use a sequence of increasing simple function $f_n$ to approximate $f^*$. Each $f_n$ can be extended to a measurable $\tilde{f}_n$ from $X$ to $[0,1]$. Then $g_n=\sup_{i\leq n}\tilde{f}_i$ is increasing and converges to a measurable function which is an extension of $f^*$.
However, the extension may exceed the range of original $f^*$. So its only a weaker result.
It also want know the proof of original problem. And where did you meet the problem? In descriptive set theory? I want to go deeper in the Polish space and measure theory. Could you provide some references?
