GLOBAL DENSITY THEOREM FOR A FEDERER MEASURE

Abstract

The local and global density theorems for the Lebesgue measure in a Euclidean space play a fundamental role in calculus. On the other hand Federer [5] proved a local density theorem for a measure with a doubling condition on a metric space.
The aim of this paper is to prove a global density theorem for a measure with a doubling condition and a class of integrable functions on a metric space. As a special case this theorem also gives a simple and constructive proof to Federer's local density theorem.
A typical example of the above measures is the Hausdorff measure on a self-similar set.