Consider an abstract Banach lattice. Under some mild assumptions, it can be identified with a Banach ideal of integrable functions with respect to a (non necessarily $\sigma$-finite) vector measure on a $\delta$-ring. Extending some nowadays well-known results for the Koml\'os property involving Cesaro sums, we prove that the weak $\sigma$-Fatou property for a Banach lattice of integrable functions $E$ is equivalent to the existence for each norm bounded sequence $(f_n)$ in $E$ of a regular method of summability $D$ such that the sequence $(f_n^D)$ converges.
"Regular methods of summability and the weak $\sigma$-Fatou property in abstract Banach lattices of integrable functions." Bull. Belg. Math. Soc. Simon Stevin 25 (4) 545 - 553, december 2018. https://doi.org/10.36045/bbms/1546570909