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A lemma on maximal sets and the theorem of Denjoy-Vitali
Published online by Cambridge University Press: 09 April 2009
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By an ∮-related family ∮ we mean a non-empty family ∮ of elements such that to each element F ∈ ∮ is associated a set R(F) of elements of ∮, called the R-class of F, which contains F. An element G ∈ R(F) is said to be R-related to F. By an R-section S of ∮ we mean a set of elements of ∮ such that for any elements F1, F2 of S either F1 ∈ R(F2) or F2 ∈R(F1). If R(F) = {F} for each F ∈ ∮ then the only R-Sections are the sets {F}. The interesting applications of the lemma proved below are to those cases when there exist R-sections which do not contain a finite number of elements.
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- Research Article
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- Copyright © Australian Mathematical Society 1964
References
[1]Denjoy, A., Une extension du théorème de Vitali, Amer. J. Math., 73 (1951), 314–56.CrossRefGoogle Scholar
[2]Finch, P. D., Integration of real-valued set functions in Abstract spaces, this Journal 4 (1964), 202–213.Google Scholar
[3]Trjitzinsky, W. J., Théorie métrique dans les espaces où il y a une mesure, Mém. des Sci. Math., CXLIII, Paris, Gauthier-Villars 1960.Google Scholar
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