On density points of subsets of metric space with respect to the measure given by Radon-Nikodým derivative (Q1978911)

Let \(X\) be a~metric space, \(\mu \) and \(\nu \) Borel measures finite on balls such that \(\mu \)-null sets and \(\nu \)-null sets coincide. Is it true that density points of a~set with respect to \(\mu \) are the same as with respect to \(\nu \)? (As a~motivation to this question, the author recalls that density points of sets on the real line can be defined without the Lebesgue measure itself using the notion of null sets only.) For measures \(\mu \) and \(\nu \) as above, the following result is established: Let \(d\mu /d\nu \) be essentially bounded and essentially bounded away from zero on every ball. Then, for any measurable set, the sets of density points with respect to \(\mu \) and \(\nu \) are identical.