On Completeness Generated by Convergence with Respect to a σ-Ideal

Abstract

We consider convergence (introduced by E. Wagner in 1981) with respect to a $\sigma$-ideal of $\mathscr{S}$-measurable real valued functions on $Y$ where $\mathscr{S}\subset \mathscr{P} (Y)$ is a $\sigma$-algebra containing a given $\sigma$-ideal $\mathscr{J}$. We check which operations preserve completeness generated by convergence with respect to a $\sigma$-ideal. We introduce uniform kinds of $\mathscr{J}$-convergence and $\mathscr{J}$-completeness and use them in a statement concerning the Fubini product of two $\sigma$-ideals.