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!