Egorov ideals

We study Egorov ideals, that is ideals on $\omega$ for which the Egorov’s theorem for ideal versions of pointwise and uniform convergences holds. We show that a non-pathological ${\mathbf{\Sigma }}_{\mathbf{2}}^{\mathbf{0}}$ ideal is Egorov if and only if it is countably generated. In particular, up to isomorphism, there are only three non-pathological ${\mathbf{\Sigma }}_{\mathbf{2}}^{\mathbf{0}}$ Egorov ideals. On the other hand, we construct $2^{\omega }$ pairwise non-isomorphic Borel Egorov ideals. Moreover, we characterize when a product of ideals is Egorov.