{"title":"广义Cantor集的Hausdorff测度的评价","authors":"K. Hatano","doi":"10.32917/HMJ/1206138659","DOIUrl":null,"url":null,"abstract":"The problem how a Hausdorff measure of a product set AxB is related to Hausdorff measures of A and B is not completely solved. This problem was first investigated by F. Hausdorff himself Q3] and later by A. S. Besicovitch and P. A. P. Moran [1], J. M. Marstrand \\ΊΓ\\ and others. Their works and investigations of similar problem for capacity (e.g. [6], [7]) show that evaluation of Hausdorff measures of generalized Cantor sets supplies many clues to this problem. In this paper we first evaluate the α-Hausdorff measure of generalized Cantor sets in the Euclidean space R. As a concequence we see the existence of a compact set in R which has infinite α-Hausdorff measure but zero incapacity (0<a<n). Next we estimate Hausdorff measures of product sets of one-dimensional generalized Cantor sets and then give examples which show that in case the α-Hausdorff measure of £Ί is infinite and the ^-Hausdorff measure of E2 is zero, the (a + β)-Hausdorff measure of Ex x E2 may either be zero, positive finite or infinite. Also these examples answer M. Ohtsuka's question in [_7J (p. 114) in the negative. The author wishes to express his deepest gratitude to Professor M. Ohtsuka for his suggesting the problem and his valuable comments.","PeriodicalId":17080,"journal":{"name":"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry","volume":"58 1","pages":"371-379"},"PeriodicalIF":0.0000,"publicationDate":"1968-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Evaluation of Hausdorff measures of generalized Cantor sets\",\"authors\":\"K. Hatano\",\"doi\":\"10.32917/HMJ/1206138659\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The problem how a Hausdorff measure of a product set AxB is related to Hausdorff measures of A and B is not completely solved. This problem was first investigated by F. Hausdorff himself Q3] and later by A. S. Besicovitch and P. A. P. Moran [1], J. M. Marstrand \\\\ΊΓ\\\\ and others. Their works and investigations of similar problem for capacity (e.g. [6], [7]) show that evaluation of Hausdorff measures of generalized Cantor sets supplies many clues to this problem. In this paper we first evaluate the α-Hausdorff measure of generalized Cantor sets in the Euclidean space R. As a concequence we see the existence of a compact set in R which has infinite α-Hausdorff measure but zero incapacity (0<a<n). Next we estimate Hausdorff measures of product sets of one-dimensional generalized Cantor sets and then give examples which show that in case the α-Hausdorff measure of £Ί is infinite and the ^-Hausdorff measure of E2 is zero, the (a + β)-Hausdorff measure of Ex x E2 may either be zero, positive finite or infinite. Also these examples answer M. Ohtsuka's question in [_7J (p. 114) in the negative. The author wishes to express his deepest gratitude to Professor M. Ohtsuka for his suggesting the problem and his valuable comments.\",\"PeriodicalId\":17080,\"journal\":{\"name\":\"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry\",\"volume\":\"58 1\",\"pages\":\"371-379\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1968-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.32917/HMJ/1206138659\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of science of the Hiroshima University Ser. A Mathematics, physics, chemistry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.32917/HMJ/1206138659","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 3
摘要
积集AxB的Hausdorff测度与a和B的Hausdorff测度之间的关系问题还没有完全解决。这个问题最初是由F. Hausdorff本人[3]，后来由A. S. Besicovitch和P. A. P. Moran [1]， J. M. Marstrand \ΊΓ\等人研究的。他们的工作和对类似容量问题的研究(例如[6]，[7])表明，广义康托集的Hausdorff测度的评价为该问题提供了许多线索。本文首先求出了欧几里德空间R中广义Cantor集的α-Hausdorff测度，得到了R中存在一个紧集，它具有无穷个α-Hausdorff测度而无容(0本文章由计算机程序翻译，如有差异，请以英文原文为准。
Evaluation of Hausdorff measures of generalized Cantor sets
The problem how a Hausdorff measure of a product set AxB is related to Hausdorff measures of A and B is not completely solved. This problem was first investigated by F. Hausdorff himself Q3] and later by A. S. Besicovitch and P. A. P. Moran [1], J. M. Marstrand \ΊΓ\ and others. Their works and investigations of similar problem for capacity (e.g. [6], [7]) show that evaluation of Hausdorff measures of generalized Cantor sets supplies many clues to this problem. In this paper we first evaluate the α-Hausdorff measure of generalized Cantor sets in the Euclidean space R. As a concequence we see the existence of a compact set in R which has infinite α-Hausdorff measure but zero incapacity (0
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