Non-additive measure theory in Riesz spaces with certain smoothenss conditions

1. Egoroff定理对任何Riesz空间值非可加测度都是有效的，只要Riesz空间具有渐近Egoroff性质。这一性质对于许多具体的Riesz空间，如任意非空集上的所有真实的函数空间和所有Lebesgue可测函数空间及其理想都是成立的. Egoroff定理对于任意Riesz空间值非可加测度都是连续的，并且具有一种称为“性质”的连续形式，只要Riesz空间具有Egoroff性质。这个版本的Egoroff定理也适用于任何具有一致自连续性，强序连续性和下连续性的非可加测度，只要假设弱o-分布性弱于Egoroff性质.引入了一个光滑性条件（重Egoroff性质），并将其应用于Riesz空间，证明了任意度量空间上的弱零可加Riesz空间值模糊Borel测度都是正则的.同时证明了Lusin定理对这类Riesz空间值非可加测度仍然成立.引入了o-光滑可数次赋范Riesz空间类，证明了经典Riesz定理对任意Riesz空间值非可加测度是自上连续和自下连续的.关于Riesz空间中紧非可加测度的Alexandroff定理在以下两种情况下仍然成立：一种是测度自连续且Riesz空间具有弱渐近Egoroff性质的情况，另一种是测度一致自连续且Riesz空间弱超分配的情况.讨论了非可加测度的正则性与连续性之间的密切联系。

1. The Egoroff theorem remains valid for any Riesz space-valued non-additive measure that 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 the space of all real functions on an arbitrary non-empty set and the space of all lebesgue measurable functions, and their ideals.2. The Egoroff theorem remains valid for any Riesz space-valued non-additive measure that is strung order continuous and possesses a form of continuity called "property (Sr 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 o-distributivity that is weaker than the Egoroffproperty.3. A smoothness condition (the multiple Egoroff property) is introduced and imposed on a Riesz space to show that every weakly null-additive Riesz space-valued fuzzy Borel measure on any metric space is regular. It is also proved that Lusin's theorem remains valid for such Riesz space-valued non-additive measures.4. The class of o-smooth countably subnormed Riesz spaces is introduced to show that the classical Riesz theorem holds for any Riesz space-valued non-additive measure that is autocontinuous from above and continuous from below.5. The Alexandroff theorem for a compact non-additive measure with values in a Riesz space is still valid for the following two cases: one is the rase that the measure is autocontinous and the Riesz space has the weak aysmptotic Egoroff property and the other is the rase that the measure is uniformly autocontinuous and the Riesz space is weakly crdistributive. A close connection between regularity and continuity of non-additive measures is also discussed.