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

具有一定平滑条件的Riesz空间中的非可加测度论

基本信息

批准号：
18540166
负责人：
KAWABE Jun
金额：
$ 2.57万
依托单位：
Shinshu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540166/
关键词：
non-additive measure asymptotic Egoroff property multiple Egoroff property Riesz space Egoroff theorem Lusin theorem Alexandroff theorem Choquet integral Radon total o-continuity statistical manifold affine connection

项目摘要

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.

1。通过假设Riesz空间具有渐近Egoroff特性，Egoroff定理对于任何Riesz空间值的非加性措施仍然有效。对于许多具体的Riesz空间，该属性都可以满足此属性，例如任意非空置集中所有实际功能的空间以及所有Lebesgue可测量功能的空间及其理想的空间。2。 eGoroff定理对于任何riesz空间值的非添加措施仍然有效，该措施是连续串行的，并具有一种称为“属性的连续性”形式，每当Riesz空间具有Egoroff财产时，在文献中具有EGOROFF的财产。此版本的Egoroff Theorem的本版本也与任何非原始级别的defition and cants the Egroff TheoRem均具有强度的效率，以下是较强的依据。与EgoroffProperty更弱的o。3。措施4。对于以下两种情况，Alexandroff定理具有与Riesz空间中值的紧凑型非加性测量值：一种是该度量是自发性的，Riesz Space具有弱的Aysmptotic Egoroff财产，另一个是Rase，该措施是均匀的自动连续性和Riess spaceisters crdistribs spaceisterib spacesterib spaceisters spacestrentib spacesterib spaceisters spaceSterib spaceSterib。还讨论了非加性措施的规律性和连续性之间的密切联系。