| Relations among gauge functions, metrics and Hausdorff measures |
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| 作者姓名： | WEN Shengyou WEN Zhiying |
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| 作者单位： | Nonlinear Scientific Center, Wuhan University, Wuhan 430072, China,Department of Mathematics, Tsinghua University, Beijing 100084, China |
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| 基金项目： | Supported by the Major State Basic Research Development Program of China (Grant No. G2000077307) |
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| 摘 要： | In this paper we discuss the relations among the doubling condition, equivalence of the Hausdorff measures and equivalence of metrics. We will show that H ρ,g1 and H ρ,g2 are equivalent for any compact metric space (X,ρ) if and only if g1 and g2 are equivalent gauge functions. Then, we prove that for given c∈(0,∞)＼{1}, H ρ,g and H cρ,g are equivalent for any compact metric space (X,ρ) if and only if the gauge function g satisfies the doubling condition, where H ρ,g is the Hausdorff measure with respect to the metric ρ and gauge function g.
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| 关 键 词： | Hausdorff measure metrics gauge function doubling condition |
Relations among gauge functions, metrics and Hausdorff measures |
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| WEN Shengyou,WEN Zhiying.Relations among gauge functions, metrics and Hausdorff measures[J].Progress in Natural Science,2003,13(4):254-258. |
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| Authors: | WEN Shengyou WEN Zhiying |
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| Institution: | 1. Nonlinear Scientific Center, Wuhan University, Wuhan 430072, China
2. Department of Mathematics, Tsinghua University, Beijing 100084, China |
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| Abstract: | In this paper we discuss the relations among the doubling condition, equivalence of the Hausdorff measures and equivalence of metrics. We will show that H ρ,g1 and H ρ,g2 are equivalent for any compact metric space (X,ρ) if and only if g1 and g2 are equivalent gauge functions. Then, we prove that for given c∈(0,∞)＼{1}, H ρ,g and H cρ,g are equivalent for any compact metric space (X,ρ) if and only if the gauge function g satisfies the doubling condition, where H ρ,g is the Hausdorff measure with respect to the metric ρ and gauge function g. |
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| Keywords: | Hausdorff measure metrics gauge function doubling condition |
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