M. Studeny: On the differentiation theorem in metric groups. Commentationes Mathematicae Universitatis Carolinae 24 (1983), n. 2, pp. 223-232.

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)

differentiation theorem
compact metric group
translation invariant metric
almost uniform measure

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The paper builds on the following works: