TITLE: Is there a Borel subset of $ \mathbb{R}^{2} $, with finite vertical cross-sections, whose projection onto the first component is non-Borel?
QUESTION [14 upvotes]: This question is related to another one that I asked two days ago.

Question. Does there exist a Borel subset $ M $ of $ \mathbb{R}^{2} $ with
    the following two properties?

The projection of $ M $ onto the first component is a non-Borel analytic
  subset of $ \mathbb{R} $.
Every vertical cross-section of $ M $ is finite, i.e., the set
  $ \{ y \in \mathbb{R} \mid (x,y) \in M \} $ is finite for every
  $ x \in \mathbb{R} $.


An affirmative answer to this question will provide a counterexample in the topic of measure theory, as explained in the linked post. It therefore holds some importance.
Thank you very much for your help!

REPLY [4 votes]: There is a pure recursion theoretical proof of the result.  The idea is as follows: By Spector-Gandy theorem, a lightface Borel set $(x,y)$ is an r.e set over $L_{\omega_1^{CK}}[x,y]$. If there are at most countably many $y$'s corresponded to the $x$ for every $x$, then it can be reduced to $L_{\omega_1^{CK}}[x]$. Then the projection to the first section becomes a $\Pi^1_1$ statement.