Doubling measures on uniform Cantor sets

TL;DR

This paper characterizes exactly which probability measures are doubling on uniform Cantor sets and explores conditions for extending these measures to the entire interval [0, 1].

Contribution

It provides a complete description of doubling measures on uniform Cantor sets and investigates their extendability to [0, 1].

Findings

Complete characterization of doubling measures on uniform Cantor sets.

Conditions identified for extending measures to the full interval.

Insights into measure extension problems for fractal sets.

Abstract

We obtain a complete description for a probability measure to be doubling on an arbitrarily given uniform Cantor set. The question of which doubling measures on such a Cantor set can be extended to a doubling measure on [0; 1] is also considered.