Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-03T19:24:36.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Capacité des Ensembles Produits

Published online by Cambridge University Press:  22 January 2016

Makoto Ohtsuka
Makoto Ohtsuka*
Affiliation:
Institut de Mathématiques, Université de Nagoya
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Si X est un ensemble mesurable au sens de Lebesgue dans un espace euclidien En et Y est un ensemble mesurable au sens de Lebesgue dans un autre espace euclidien Em, alors X × Y est mesurable dans En+m= Enx Em et sa mesure est égale au produit de la mesure de X et celle de Y. Mais il n’existe pas de telle relation simple ni pour la capacité d’ordre général ni pour la mesure de Hausdorff de dimension générale. Dans le présent mémoire nous essayons d’évaluer la capacité des ensembles produits au moyen de la capacité ou de la mesure de Hausdorff des ensembles composants.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 12 , December 1957 , pp. 95 - 130
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1957

References

Bibliographie

[ 1 ] Aronszajn, N. and Smith, K. T.: Functional spaces and functional completion, Ann. Inst. Fourier, 6 (1956), pp. 125–185.CrossRefGoogle Scholar
[ 2 ] Besicovitch, A. S.: Concentrated and rarefied sets of points, Acta Math., 62 (1933), pp. 289–300.CrossRefGoogle Scholar
[ 3 ] Besicovitch, A. S.: On existence of subsets of finite measure of sets of infinite measure, Indag. Math., 14 (1952), pp. 339–344.CrossRefGoogle Scholar
[ 4 ] Besicovitch, A. S. and Moran, P. A. P.: The measure of product and cylinder sets, J. Lond. Math. Soc., 20 (1945), pp. 110–120.Google Scholar
[ 5 ] Bogoliouboff, N. et Kryloff, N.: La théorie générale de la mesure dans son application â l’étude des systèmes dynamiques de la mécanique non linéaire, Ann. Math., 38 (1937), pp. 65–113.Google Scholar
[ 6 ] Carleson, L.: On a class of meromorphic functions and its associated exceptional sets, Thèse, Uppsala (1950).Google Scholar
[ 7 ] Choquet, G.: Theory of capacities, Ann. Inst. Fourier, 5 (1955), pp. 131–295.CrossRefGoogle Scholar
[ 8 ] Choquet, G.: Les noyaux réguliers en théorie du potentiel, C. R. Acad. Sci. Paris, 243 (1956), pp. 635–638.Google Scholar
[ 9 ] Davies, R. O.: Subsets of finite measure in analytic sets, Indag. Math., 14 (1952), pp. 488–489.CrossRefGoogle Scholar
[10] Deny, J. et Lelong, P.: Etude des fonctions sousharmoniques dans un cylindre ou dans un cône, Bull. Soc. Math. France, 75 (1947), pp. 89–112.CrossRefGoogle Scholar
[11] Eggleston, H. G.: The Besicovitch dimension of cartesian product sets, Proc. Camb. Phil. Soc, 46 (1950), pp. 383–386.CrossRefGoogle Scholar
[12] Eggleston, H. G.: A correction to a paper on the dimension of cartesian product sets, ibid., 49 (1953), pp. 437–440.Google Scholar
[13] Federer, H.: Some integralgeometric theorems, Trans. Amer. Math. Soc, 77 (1954), pp. 238–261.CrossRefGoogle Scholar
[14] Freilich, G.: On the measure of cartesian product sets, ibid., 69 (1950), pp. 232–275.Google Scholar
[15] Freilich, G.: Two-dimensional measure in 3-space, Proc. Amer. Math. Soc, 6 (1955), pp. 631–633.CrossRefGoogle Scholar
[16] Hausdorff, F.: Dimension und ausseres Mass, Math. Ann., 79 (1919), pp. 157–179.CrossRefGoogle Scholar
[17] Kametani, S.: Positive definite integral quadratic forms and generalized potentials, Proc. Imp. Acad. Japan, 20 (1944), pp. 7–14.Google Scholar
[18] Kametani, S.: On Hausdorff’s measure and generalized capacities with some of their applications to the theory of functions, Jap. J. Math., 19 (1945), pp. 217–257.CrossRefGoogle Scholar
[19] Kametani, S.: Progrès recent de la théorie du potentiel, Mathématiques modernes I, Tokyo (1950), pp. 62–105, (en japonais).Google Scholar
[20] Kametani, S.: A note on a metric property of capacity, Natur. Sci. Rep. Ochanomizu Univ., 4 (1953), pp. 51–54.Google Scholar
[21] Marstrand, J. M.: The dimension of Cartesian product sets, Proc. Camb. Phil. Soc, 50 (1954), pp. 198–202.CrossRefGoogle Scholar
[22] Moran, P. A. P.: Additive functions of intervals and Hausdorff measure, ibid., 42 (1946), pp. 15–23.Google Scholar
[23] Moran, P. A. P.: On plane sets of fractional dimensions, Proc. Lond. Math. Soc, 51 (1949), pp. 415–423.CrossRefGoogle Scholar
[24] Ohtsuka, M.: Capacité d’ensembles de Cantor généralisés, Nagoya Math. J., 11 (1957), pp. 151–160.CrossRefGoogle Scholar
[25] Randolph, J.: On generalizations of length and area, Bull. Amer. Math. Soc, 42 (1936), pp. 268–274.CrossRefGoogle Scholar