On Density Points of Subsets of Metric Space with Respect to the Measure Given by Radon-Nikodym Derivative

Abstract

A natural problem is: what can we say about density points of sets with respect to measure \(\nu\), which has the same \(\sigma\)- ideal of null sets as another measure \(\mu\). We will show that, in general, the density points of \(A\) with respect to \(\mu\) need not be the density points of \(A\) with respect to \(\nu\), but we will find some sufficient conditions for \(\nu\) under which density points for \(\mu\) are also density points for \(\nu\).