Abstract
Within an abstract theory of point sets the author has successfully unified a substantial number of the analogous theorems concerning Lebesgue measure and Baire category. It has been shown that the Lebesgue measurable sets coincide with the sets having the abstract Baire property with respect to the family of all closed sets of positive Lebesgue measure and the question was raised in [2] whether the sets measurable with respect to certain Hausdorff measures were the same as the sets having the abstract Baire property with respect to the family of all closed sets of positive Hausdorff measure. In this article we establish a general theorem which, under the assumption of the continuum hypothesis, gives an affirmative answer to this question.
Similar content being viewed by others
References
Besicovitch A.S.,Concentrated and rarified sets of points, Acta Mathematica,62 (1934), 289–300.
Morgan J.C. II,Infinite games and singular sets, Colloquium Mathematicum,29 (1974), 7–17.
Morgan J.C. II,Baire category from an abstract viewpoint, Fundamenta Mathematicae,94 (1977), 13–23.
Morgan J.C. II,On product bases, Pacific Journal of Mathematics99 (1982), 105–126.
Rogers C.A.,Hausdorff Measures, Cambridge 1970.
Sierpinski W.,Hypothése du Continu, New York 1956.
Szpilrajn E., (Marczewski),Sur un ensemble non mesurable de M. Sierpinski. Sprawozdania z posiedzén Towarzystwa Naukowego Warszawskiego, Wydzial III (Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III),24 (1931), 78–85.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Morgan, J.C. Measurability and the abstract Baire property. Rend. Circ. Mat. Palermo 34, 234–244 (1985). https://doi.org/10.1007/BF02850698
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02850698