The Egoroff theorem for non-additive measures in Riesz spaces

The Egoroff theorem remains valid for any Riesz space-valued non-additive measure which is continuous from above and below by assuming that the Riesz space has the asymptotic Egoroff property. This property is satisfied for many concrete Riesz spaces, such as RS of all real functions on a non-empty set S, L0[0,1] of all Lebesgue measurable functions, and their ideals.