Dependence Of Hausdorff Measure And Dimension With The Metric Of Underlying Space

This paper discussed the construction of metric space on the nested geometrical object. Given a nested geometrical object K in R~n and a continuous gauge function h(t), a new metricρwas constructed on K such that 0＜(?)~h(K)＜+∞in the new metric space(K,ρ).Particularly,if the gauge function is h(t)=t~s,then for any positive finite number s,it's also possible to construct a new metricρon K such that(?)~s(K)=1 and dim _PK=dim _BK=dim _HK=s.If K is a self similar set which satisfies strong separated condition,then for any positive finite number s,we can construct a new metricρon K,such that K is also a self similar set and 0＜(?)~h(K)＜+∞.