On completeness generated by convergence with respect to a \(\sigma\)-ideal (Q2464613)

Let \(\mathcal S\) be a \(\sigma\)-algebra contained in the power set \({\mathcal P} (Y)\) of \(Y\), \(\mathcal I\) a \(\sigma\)-ideal in \({\mathcal P} (Y)\) and \(\mathcal F(S)\) the space of real-valued \(\mathcal S\)-measurable functions on \(Y\). A sequence \(f_n\) in \(\mathcal F(S)\) converges \(\mathcal I\)-a.e. to \(f\in\mathcal F(S)\) if the set of \(y\in Y\) for which \(f_n(y)\) does not converge to \(f(y)\) belongs to \(\mathcal I\); the sequence \(f_n\) is said to converge to \(f\) with respect to \(\mathcal I\) if each subsequence of \(f_n\) has a subsequence convergent \(\mathcal I\)-a.e. to \(f\) [see \textit{E. Wagner}, Fund. Math. 112, 89--102 (1981; Zbl 0386.28005)]. \(f_n\) is said to be \(\mathcal I\)-Cauchy if the difference of any two subsequences of \(f_n\) converges to \(0\) with respect to \(\mathcal I\). \(\mathcal F(S)\) is \(\mathcal I\)-complete if every \(\mathcal I\)-Cauchy sequence converges in \(\mathcal F(S)\) with respect to \(\mathcal I\). The author studies which operations of \(\sigma\)-algebras/\(\sigma\)-ideals preserve completeness. A typical result says: For \(n\in \mathbb{N}\), let \({\mathcal S}_n\) be a \(\sigma\)-algebra contained in \({\mathcal P} (Y)\) and \({\mathcal I}_n\) a \(\sigma\)-ideal in \({\mathcal P} (Y)\) such that \({\mathcal I}_n\) has a base in \({\mathcal S}_n\) and \({\mathcal F(}{\mathcal S}_n)\) is \({\mathcal I}_n\)-complete; let \({\mathcal S} =\bigcap _{n=1}^\infty {\mathcal S}_n\) and \({\mathcal I} =\bigcap _{n=1}^\infty {\mathcal I}_n\). Then \(\mathcal F(S)\) is \(\mathcal I\)-complete. The author also studies a uniform kind of \(\mathcal I\)-convergence and \(\mathcal I\)-completeness and uses them in a statement concerning the Fubini product of two \(\sigma\)-ideals.