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