Weight Functions And Metrics On Compact Metrizable Spaces

The nested partition of a compact metric space plays an important role in many areas of mathematics,such as the binary division of the unit interval[0,1]and the construction of trisection Cantor set and so on.This construction process is mainly to divide a space into a finite number of subsets,then divide each subset into finite pieces,and repeat the process.Given a partition,we hope to describe the "size" of these subsets for different pieces.Thus we introduce the concept of weight function,i.e.giving each pieces a value between 0 and 1.Given a metric d on compact metric space,the weight function can be induced naturally.However,this paper is interested in studying the problem from the opposite direction,that is,given a weight function g,there exist a corresponding metric d under what conditions.This is a report of recent work of J.Kigami[10]on this topic.