2
$\begingroup$

Is it true the following statement?

Given two Polish spaces $X,Y$ and a Borel function $f:X\rightarrow Y$, there exists a Polish space $Z$ and a Borel function $g:X \rightarrow Y\times Z$ with closed graph such that $f(x) \ = \pi_Y(g(x))$ for all $x \in X$.

In case it was true, do you have any hint for the proof?
Thanks!

$\endgroup$

1 Answer 1

6
$\begingroup$

Yes, this is true: by Exercise 13.5 in Kechris' "Classical Descriptive Set Theory", for any Borel function $f:X\to Y$ between Polish spaces there exists a continuous bijective map $i:Z\to X$ from a Polish space $Z$ such that the function $f\circ i:Z\to Y$ is continuous.

Now consider the function $g:X\to Y\times Z$, $g:x\mapsto (f(x),i^{-1}(x))$ and observe that it has closed graph $$\Gamma=\{(x,y,z)\in X\times Y\times Z:y=f(x),\; x=i(z)\}=\{(x,y,z)\in X:y=f\circ i(z),\;x=i(z)\}$$by the continuity of the functions $i$ and $f\circ i$. It is clear that $f=\pi_Y\circ g$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.