The Egoroff property and the Egoroff theorem in Riesz space-valued non-additive measure theory

The Egoroff theorem remains valid for any Riesz space-valued non-additive measure which is strong order continuous and possesses a form of continuity called “property (S)” in the literature, whenever the Riesz space has the Egoroff property. This version of the Egoroff theorem is also valid for any non-additive measure with the property of uniform autocontinuity, strong order continuity and continuity from below by assuming only the weak σ-distributivity which is weaker than the Egoroff property.