利用非标准分析的方法, 给出了S0-类实函数的一个积分不等式和一个积分等式: (1)设 A是标准的完备度量空间 (X,d) 中的一个准标准内的子集, μ是 X上的一个Borel正则有限的标准外测度. 若 f 是 X上的一个非负的S0-类实函数, 则有°(∫Afdμ)≤∫°A°fdμ特殊情况当f≡1时, 有°(μ(a))≤μ(°A);(2)设E 是标准的 s-紧集, 若 f 是 E 上的一个 S0-类实函数, 则°(∫EfdHs)=∫E°fdHs, 这里 °(*)指的是 * 的影子.并给出了这些结果在分形几何中用以判断一个分形集其内部是否非空的方法及一个分形函数在 Hausdorff 测度空间上的积分的计算方法,并给出了相应的实例加以验证.
Abstract
In this paper, utilizing the method of nonstandard analysis, we prove that: (1) for any near standard interior subset A and a standard Borel regular finite outer measure μ of standard complete metric space (X,d), if f is a nonnegative real function in S0(X), then the integral inequality °(∫Afdμ)≤∫°A°fdμ holds, in special case °(μ(a))≤μ(°A) for f≡1; (2) if f ∈ S0(X) and E is a s-compact subset of X, then °(∫EfdHs)=∫E°fdHs, where ?(*) denote the shadow of *. Moreover, by using these results we give a judgment method of fractal with the interior nonempty and a calculating method for integration with respect to a fractal function on the Hausdorff measure space, the efficiency of the two methods is shown with examples.
关键词
非标准分析 /
测度 /
Hausdorff度量 /
分形
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Key words
nonstandard analysis /
measure /
Hausdorff metric /
fractal
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参考文献
[1] Robinson A., Nonstandard Ananlysis, North-Holland Publishing company, Amsterdam, 1974.
[2] Nelson E., Internal theory, Bull Amer Math., 1977, 83: 1165-1198.
[3] Diener F., Reeb G., Analyse Non Standard, Hermann Editeurs des Sciences et des Arts, Paris, 1989.
[4] Halmos P. R., Measure Theory-Graduate Texts in Mathemetics, Springer-Verlag, Berlin, (v.18), 1970.
[5] Falconer K. J., Fractal Geometry-Mathematical Foundations and Applications, John Wiley, New York, 1990.
[6] Bedford T., Kamae T., Stieltjes integration and stochastic calculus with respect to Self-affine functions, Japan J. Indust. Appl. Math., 1991, 8: 445-459.
[7] McMullen C., The Hausdorff dimension of general sierpiíski carpets, Nagoya Math.J., 1984, 96: 1-9.
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脚注
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基金
海南省教育厅自然科学资助项目及海南大学211专项资助项目
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