Borel images of sets of reals (Q1898961)

This paper generalizes several results about cardinal invariants less than the continuum by considering Borel images. A set \(X\subseteq 2^\omega\) is an \(\text{SR}^{\mathcal M}\) set iff for every Borel \(H\subseteq 2^\omega \times 2^\omega\) such that \(H_x\) is meager \(({\mathcal M})\) for every \(x\in 2^\omega\), \(\bigcup_{x\in X} H_x\) is meager \(({\mathcal M})\). It is shown that a set \(X\subseteq 2^\omega\) is an \(\text{SR}^{\mathcal M}\) set iff \(Y+F \in{\mathcal M}\) for every Borel image \(Y\) of \(X\) and \(F\in {\mathcal M}\), and every Borel image of \(X\) in \(\omega^\omega\) is covered by a countable union of compact sets. The same result is proved if the ideal of Lebesgue measure zero sets \({\mathcal N}\) replaces the ideal of meager sets \({\mathcal M}\).