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

1.假设Riesz空间具有渐近Egoroff性质，则Egoroff定理对任何自上而下连续的Riesz空间值非可加测度仍然有效。这一性质对许多具体的Riesz空间都是满足的，例如任意非空集上的所有实函数的空间和所有Lebegue可测函数的空间，以及它们的理想。当Riesz空间具有Egoroff性质时，Egoroff定理仍然适用于任何Riesz空间值非可加测度，该测度是串顺序连续的，并且具有一种形式的连续性，在文献中称为“性质(Sr)”。这一版本的Egoroff定理仅假设弱o-分配性弱于Egoroff性质，对任何具有一致自连续、强序连续和自下而上连续性质的非加性测度也是有效的。在Riesz空间上引入了一个光滑性条件(多重Egoroff性质)，证明了任意度量空间上的每一个弱零可加的Riesz空间值模糊Borel测度都是正则的。并证明了Lusin定理对于这类Riesz空间值非可加测度仍然有效。引入O-光滑可数次范数Riesz空间类，证明了经典Riesz定理对任意自上而下连续的Riesz空间值非可加测度成立。值在Riesz空间中的紧非可加测度的Alexandroff定理在以下两种情况下仍然有效：一是该测度是自连续的，且Riesz空间具有弱属性Egoroff性质；二是该测度是一致自连续的，且Riesz空间是弱分配的.还讨论了非加性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.