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  1. 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 Hardy-Littlewood maximal inequality and that needs the ball-doubling property of the metric-measure 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 metric-measure space or not.

  2. 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. Springer-Verlag, New York, 2001. x+140 pp.

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