This might be a very easy question, and it might be better for mathstackexchange in which case I apologize. I'm stuck on something an anonymous referee wrote to me about a paper of mine and I'm hoping for some clarity.


Suppose $X$ and $Y$ are Polish spaces and $A \subseteq X \times Y$ is Borel. It's well known that if for each $y \in Y$, $A\_y =\{x \; | \; (x, y) \in A\}$ is "small" in various sense, then in fact $proj\_X(A)$ is Borel. For instance, in Kechris' Classical Descriptive Set Theory Theorem 35.46 proves that if $A$ has $\mathcal K\_\sigma$ sections then $proj\_X(A)$ is Borel. My question is simply, what if the sections are null for (some appropriate version of) Lebesgue measure?


For the specific case I'm interested in, let $\mu$ denote (any choice of formulating) the Lebesgue measure on $\omega^\omega$ and let $f:\omega^\omega \to \omega^\omega$ be a partial Borel function with $\mu$-null domain. Is the domain of $f$ Borel?


Thanks!



Do Borel subsets of the plane with null sections have Borel projections?
Notice that $\omega^\omega$ can be embedded to a null subset of itself by sending any sequence $(a\_0,a\_1,a\_2,\dots)$ to $(a\_0,0,a\_1,0,a\_2,0,\dots)$. So any Borel phenomenon that can happen in $\omega^\omega$ can also happen in a null subset.


