关于度量及纲函数的Hausdorff测度的两个反例

TWO COUNTEREXAMPLES ABOUT HAUSDORFF MEASURE WITH RESPECT TO METRIC AND GAUGE FUNCTION

本文研究了度量空间中Hausdorff测度与度量及纲函数的关系.利用拓扑学及Hausdorff测度论中一些性质,构造了两反例来说明存在不等价纲函数g,h和某一紧度量空间(ρ,X),使得Hρ,g与Hρ,h对此紧度量空间等价等问题.这些反例有助于从另一个角度理解文胜友、文志英[4]中主要结果.

In this paper,we investigate the relationship among Hausdorff measure,metric and gauge function.By using some properties in topology and Hausdorff measure theory,we construct two counterexamples to illustrate problems such as whether there exist two inequivalent gauge functions g,h and a compact metric space（ρ,X） such that Hρ,gand Hρ,hare equivalent with respect to（ρ,X）.These counterexamples can help us to understand the main results in Wen Shenyou and Wen Zhiying [4] better.