To formulate Lebesgue differentiation theorem one needs a metric and a measure. Apart from the Euclidean spaces i.e. $\mathbb R^d$, the theorem holds true for homogeneous groups (e.g. Heisenberg group) and spaces of homogeneous type. (E. Stein: Harmonic Analysis; S. Krantz: Panorama of Harmonic analysis). The argument needs HardyLittlewood maximal inequality and that needs the balldoubling property of the metricmeasure space. But the theorem is also true for some spaces without that property (e.g hyperbolic spaces) as proved by Statnton and Tomas (1979 Proc. AMS). I am wondering if it is known by now where such a theorem is true, is there any criterion by which we can determine whether this theorem will be true in a metricmeasure space or not.
Wikipedia says that Lebesgue Density theorem is a weaker version of Lebesgue differentiation theorem. In fact Lebesgue Density theorem is the Lebesgue differentiation theorem for indicator functions. Is there a space where the density theorem is true but not the differentiation theorem?
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A standard reference on derivation theory:
Hayes & Pauc, Derivation and Martingales (Springer 1970)
(plug) there is a chapter on derivation in:
Edgar & Sucheston, Stopping Times and Directed Processes (Cambridge Univ Pr 1992)
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MR1800917 Heinonen, Juha. Lectures on analysis on metric spaces. Universitext. SpringerVerlag, New York, 2001. x+140 pp.