Daniell-Loomis integrals (Q1265152)

An integral extension of Lebesgue power, for arbitrary nonnegative linear functionals on functions vector lattices, was discussed by \textit{P. Bobillo Guerrero} and \textit{M. Díaz Carrillo} [Arch. Math. 49, 245-256 (1987; Zbl 0612.28010); ibid. 52, No. 3, 258-264 (1989; Zbl 0674.28005)]. Here, this extension process is generalized by ``localization'', using an appropriate local convergence in measure that has also been used to prove convergence theorems. Various descriptions of the set of integrable functions are also given, and in particular, a Darboux type characterization is stated. Riemann-\(\mu\), abstract Riemann-Loomis and Bourbaki integrals are subsumed.