On a theorem in the theory of binary relations
G. Fodor (1951)
Compositio Mathematica
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G. Fodor (1951)
Compositio Mathematica
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Ihor Stasyuk, Edward D. Tymchatyn (2013)
Colloquium Mathematicae
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Spaces of finite n-dimensional Hausdorff measure are an important generalization of n-dimensional polyhedra. Continua of finite linear measure (also called continua of finite length) were first characterized by Eilenberg in 1938. It is well-known that the property of having finite linear measure is not preserved under finite unions of closed sets. Mauldin proved that if X is a compact metric space which is the union of finitely many closed sets each of which admits a σ-finite linear...
Miklós Laczkovich, Ákos K. Matszangosz (2015)
Colloquium Mathematicae
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The classical Denjoy-Young-Saks theorem gives a relation, here termed the Denjoy property, between the Dini derivatives of an arbitrary one-variable function that holds almost everywhere. Concerning the possible generalizations to higher dimensions, A. S. Besicovitch proved the following: there exists a continuous function of two variables such that at each point of a set of positive measure there exist continuum many directions, in each of which one Dini derivative is infinite...
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.
Robert Morris Pierce
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James Fickett, Jan Mycielski (1979)
Colloquium Mathematicae
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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 E. Zink (1966)
Colloquium Mathematicae
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Ladislav Mišík (2001)
Acta Mathematica et Informatica Universitatis Ostraviensis
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Noboru Endou (2015)
Formalized Mathematics
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In our previous article [22], we showed complete additivity as a condition for extension of a measure. However, this condition premised the existence of a σ-field and the measure on it. In general, the existence of the measure on σ-field is not obvious. On the other hand, the proof of existence of a measure on a semialgebra is easier than in the case of a σ-field. Therefore, in this article we define a measure (pre-measure) on a semialgebra and extend it to a measure on a σ-field. Furthermore,...
Michał Morayne (1987)
Colloquium Mathematicae
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M. Morayne (1984)
Colloquium Mathematicae
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Michał Morayne (1987)
Colloquium Mathematicae
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T. Świątkowski (1967)
Colloquium Mathematicae
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