| Project/Area Number |
18540166
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Shinshu University |
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Principal Investigator |
KAWABE Jun Shinshu University, Faculty of Engineering, Professor (50186136)
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| Co-Investigator(Kenkyū-buntansha) |
SAKAI Yuji Shinshu University, Faculty of Engineering, Professor (80021004)
KIMURA Morishige Shinshu University, Faculty of Engineering, Professor (00026345)
YAMASAKI Motohiro Shinshu University, Faculty of Engineering, Associate Professor (30021017)
TAKANO Kazuhiko Shinshu University, Professor (80252063)
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| Project Period (FY) |
2006 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥4,010,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥510,000)
Fiscal Year 2007: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2006: ¥1,800,000 (Direct Cost: ¥1,800,000)
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| Keywords | 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 |
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| Research Abstract |
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.
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