Uniqueness of the polar factorisation and projection of a vector-valued mapping
G. R. Burton, R. J. Douglas (2003)
Annales de l'I.H.P. Analyse non linéaire
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G. R. Burton, R. J. Douglas (2003)
Annales de l'I.H.P. Analyse non linéaire
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Noboru Endou (2017)
Formalized Mathematics
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The purpose of this article is to show Fubini’s theorem on measure [16], [4], [7], [15], [18]. Some theorems have the possibility of slight generalization, but we have priority to avoid the complexity of the description. First of all, for the product measure constructed in [14], we show some theorems. Then we introduce the section which plays an important role in Fubini’s theorem, and prove the relevant proposition. Finally we show Fubini’s theorem on measure.
Robert Morris Pierce
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Noboru Endou (2016)
Formalized Mathematics
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In this article we formalize in Mizar [5] product pre-measure on product sets of measurable sets. Although there are some approaches to construct product measure [22], [6], [9], [21], [25], we start it from σ-measure because existence of σ-measure on any semialgebras has been proved in [15]. In this approach, we use some theorems for integrals.
Edward Marczewski, Roman Sikorski (1951)
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Chou, So-Hsiang, Nguyen, Truc T. (1990)
International Journal of Mathematics and Mathematical Sciences
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Michael Wendt (1994)
Cahiers de Topologie et Géométrie Différentielle Catégoriques
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Robert E. Zink (1966)
Colloquium Mathematicae
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W. Orlicz (1951)
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Bo Zhang, Hiroshi Yamazaki, Yatsuka Nakamura (2006)
Formalized Mathematics
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In this article, we first discuss the relation between measure defined using extended real numbers and probability defined using real numbers. Further, we define completeness of probability, and its completion method, and also show that they coincide with those of measure.