Abstract

Davies's example of a compact metric space with two different finite Borel measures agreeing on balls can be embedded isometrically into a compact metric group with a translation invariant metric. It is shown that these measures can be always chosen mutually singular. In particular, the differentiation theorem in convergence in measure need not hold in a compact metric group. On the other hand, it is shown that on a metric space with an almost uniform measure a weak type of the differentiation theorem holds for each measure.

AMS classification 28A15 (54E50)

Keywords

differentiation theorem

compact metric group

translation invariant metric

almost uniform measure

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The paper builds on the following works:

• R. O. Davies: Measures not approximable or not specifiable by means of balls. Mathematica 18 (1971), pp. 157-160.
• P. Mattila: Differentiation of measures in uniform spaces. In Measure Theory Oberwolfach 1979, Lecture Notes in Mathematics 794, Springer-Verlag, Berlin 1980, pp. 261-284.