Rademacher theorem states that every Lipschitz function is differentiable almost everywhere with respect to the Lebesgue measure. In this talk I will explain how this statement should be modified when the Lebesgue measure is replaced by a singular measure, in particular I will show that the differentiability properties of Lipschitz functions with respect to such a measure are exactly described by its decompositions in terms of one-dimensional rectifiable measures. This result is directly related to recent work by many authors, including myself, David Bate, Marianna Csornyei, Peter Jones, Andrea Marchese, and David Preiss.
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